DETERMINATION OF HEAT CAPACITIES AT CONSTANT VOLUME FROM
EXPERIMENTAL (P-RHO-T) DATA
A Thesis
by
ANDREA DEL PILAR TIBADUIZA RINCON
Submitted to the Office of Graduate and Professional Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
Chair of Committee, Kenneth R. Hall
Co-Chair of Committee, James C. Holste
Committee Members, Ioannis Economou
Maria Barrufet
Head of Department, M. Nazmul Karim
August 2015
Major Subject: Chemical Engineering
Copyright 2015 Andrea Del Pilar Tibaduiza Rincon
ii
ABSTRACT
This work examines the uncertainty in the determination of the heat capacity at
constant volume from experimental volumetric data. The proposed methodology uses the
experimental (P-ρ-T) data for a ternary mixture of methane, ethane and propane measured
over a range of temperatures from 140 to 500 K at pressures up to 200 MPa. The resulting
approach determines truly isochoric data and calculates isochoric densities with a
maximum deviation of ± 0.5 % from the predictions of GERG 2008 EoS. Values for the
first and second derivative of the pressure with respect to temperature at constant density
needed for determining the caloric properties come from a fit of the (P-ρ-T) data using a
rational equation and also from numerical techniques. By transformation of the data, it is
possible to calculate the residual and absolute heat capacities at constant volume with a
maximum uncertainty of 1.5% that results from comparing the differences in the
predictions using analytical and numerical approaches, which reflects the uncertainty of
the experimental (P-ρ-T) data.
iii
DEDICATION
Armando, Esther, Camilo, Jackeline
and
Our Families
iv
ACKNOWLEDGMENTS
I hereby want to thank Dr. Kenneth R. Hall for his guidance and advice on this
research project. He honored me with the opportunity to join the Thermodynamics
Research Group in College Station, TX and this opened to door to develop my professional
path to obtain my Master Degree at Texas A&M University. I appreciate his patience and
understanding during the challenging times, especially during the transition to Doha,
Qatar. I am grateful to Dr. Hall for financially supporting me for the last four years and
for his support both professional and personal.
I would like to thank Dr. James Holste for his invaluable contributions and rigorous
scientific knowledge that enhanced the accuracy and applicability of this research project.
Dr. Holste was my supervisor during the experimental stage of this work and other projects
with the oil and gas industry developed in the Thermodynamics Research Group in
College Station, TX. I would like to gratefully acknowledge my advisory committee
members, Dr. Ioannis Economou and Dr. Maria Barrufet for their time, advices and
suggestions.
I had one of the most fruitful time in College Station, TX thanks to my colleagues
in our research group. I appreciate the help I received from Dr. Diego Cristancho and Dr.
Ivan Mantilla during the early stages of my lab experience. I feel very lucky to continue
my professional development working hand by hand in challenging projects with Mr.
Martin Gomez and Mr. Robert Browne in the same research group. I would like to thank
them for their cooperation, time and friendship.
v
I would like to acknowledge the guidance and support received from the faculty
members and administrative staff of Texas A&M University at Qatar. Special thanks to
Dr. Ioannis Economou, professor of the Chemical Engineering Department who besides
to integrate my Committee of graduation was a fundamental support in my professional
and personal decisions. I am very blessed and happy to have discovered and experienced
the Middle East culture and the friendship found in Qatar. I also want to acknowledge
Qatar Foundation for the financial support throughout my Master program.
Above all, I am very indebted and grateful to my parents Armando Tibaduiza and
Esther Rincon for their unconditional love, support and understanding. I send thanks to
my whole family and friends in Colombia and to my brother and sister, Camilo and
Jackeline, they are my driving force and my motivation to continue learning and growing
every day.
vi
NOMENCLATURE
n Total number of moles (mol)
Density (kg/m3)
V Volume (m3)
T Temperature (K)
P Pressure (MPa)
CP Heat capacity at constant pressure (kJ/kmol)
Cv Heat capacity at constant volume (kJ/kmol)
R Universal gas constant: 8.3144621 (kJ/kmol-K)
U Uncertainty (%)
k Ratio of heat capacities
Superscripts
* Reference condition
° Calibration condition
' True isochore condition
ig Ideal gas state
r Residual property
vii
Subscripts
c Cell
l Transmission line
t Pressure Transducer
Greek letters
α Thermal expansion coefficient (K-1)
β Isothermal compressibility coefficient (MPa-1)
Abbreviations
MSD Magnetic Suspension Densimeter
LPI Low Pressure Isochoric Apparatus
HPI High Pressure Isochoric Apparatus
EoS Equation of State
Eq Equation
OD External Diameter
ID Internal Diameter
PT30K 30,000 psi Range Pressure Transducer
Cu-Be Beryllium Cooper 175 Alloy
Iso Isochore
Rms Root mean square
viii
TABLE OF CONTENTS
Page
ABSTRACT .......................................................................................................................ii
DEDICATION ................................................................................................................. iii
ACKNOWLEDGMENTS ................................................................................................. iv
NOMENCLATURE .......................................................................................................... vi
TABLE OF CONTENTS ............................................................................................... viii
LIST OF FIGURES ............................................................................................................ x
LIST OF TABLES ......................................................................................................... xiii
1. INTRODUCTION .......................................................................................................... 1
1.1 Economic Impact of Natural Gas ............................................................................. 1
1.2. Literature Review of Calorimetric Measurements .................................................. 5 1.2.1 Adiabatic Calorimeter for Heat Capacities Measurements16 ............................ 6
1.2.2 Flow Calorimeters for Constant Pressure Heat Capacities Measurements ....... 7 1.2.3 Caloric Properties for Natural Gas Mixtures12 .................................................. 8
1.3 Research Proposal and Objectives of the Study ....................................................... 9
2. EXPERIMENTAL CHARACTERIZATION OF NATURAL GAS MIXTURES ..... 11
2.1 Magnetic Suspension Densimeter (MSD)8,9 .......................................................... 11 2.2 Low and High Pressure Isochoric Apparatus6,9,30 .................................................. 14
3. HEAT CAPACITIES FROM EXPERIMENTAL (P-RHO-T) DATA ........................ 17
3.1 Stage 1. Experimental Data Acquisition ................................................................ 17 3.2 Stage 2. Isochoric Corrections ............................................................................... 20
3.2.1 Definition of Reference State .......................................................................... 21 3.2.2 Isochoric Density Determination ..................................................................... 23 3.2.3 True Isochores Determination ......................................................................... 35
3.3 Stage 3. Derivative Determination ......................................................................... 37
3.4 Stage 4. Residual and Absolute Properties ............................................................ 55 3.5 Stage 5. Uncertainty Analysis ................................................................................ 64
ix
4. CONCLUSIONS .......................................................................................................... 71
REFERENCES ................................................................................................................. 75
APPENDIX A. EXPERIMENTAL ISOTHERMAL AND ISOCHORIC DATA .......... 79
APPENDIX B. ISOCHORIC DENSITY AND TRULY ISOCHORIC DATA .............. 89
APPENDIX C. REGRESSION RESULTS OF FIT P VS T............................................ 96
APPENDIX D. FIRST AND SECOND DERIVATIVES ............................................... 97
APPENDIX E. EXPRESSION OF RESIDUAL HEAT CAPACITY ........................... 104
APPENDIX F. REGRESSION RESULTS OF INTEGRAND FUNCTION ................ 106
APPENDIX G. IDEAL GAS HEAT CAPACITY DIPPR PROJECT .......................... 110
APPENDIX H. RESIDUAL AND ABSOLUTE HEAT CAPACITY .......................... 111
APPENDIX I. INTEGRAND FUNCTION OF NUMERICAL DERIVATIVES ......... 118
APPENDIX J. MATLAB CODE ................................................................................... 120
x
LIST OF FIGURES
Page
Figure 1. Distribution in consumption of natural gas in U.S. 20132 .................................. 2
Figure 2. Gas demand in American market3 ...................................................................... 3
Figure 3. US generation from coal, gas-fired plants and renewable energies3 .................. 3
Figure 4. (a) Basic scheme for the MSD25 (b) Operating modes of the MSD25 ............... 12
Figure 5. Schematic diagram of the isochoric experiment30 ............................................ 14
Figure 6. Isochoric cell cut view for the high pressure equipment6 ................................. 15
Figure 7. Methodology to determine (Cv) from experimental (P-ρ-T) data ..................... 17
Figure 8. Experimental data of residual gas sample ......................................................... 19
Figure 9. Noxious volume in isochoric apparatus ............................................................ 20
Figure 10. Percentage deviation of density values from MSD......................................... 22
Figure 11. Per cent deviation of isochoric densities in the LPI ........................................ 29
Figure 12 Layout of high pressure isochoric cell ............................................................. 32
Figure 13. Percentage of deviation of isochoric densities from HPI................................ 34
Figure 14. True isochores diagram ................................................................................... 36
Figure 15. Strategy to calculate numerical and analytical derivatives ............................. 37
Figure 16 Residuals in pressure for isochores 1 to 3 using Eq. 25 .................................. 42
Figure 17 Residuals in pressure for isochores 1 to 3 using Eq. 30 .................................. 42
Figure 18 Residuals in pressure for isochores 4 and 5 using Eq. 25 ................................ 43
Figure 19 Residuals in pressure for isochores 4 and 5 using Eq. 30 ................................ 43
Figure 20 Residuals in pressure for isochores 6 to 10 using Eq. 25 ................................ 44
xi
Figure 21 Residuals in pressure for isochores 6 to 10 using Eq. 30 ................................ 44
Figure 22 First derivative for isochore 1 .......................................................................... 45
Figure 23 Second derivative for isochore 1 ..................................................................... 45
Figure 24 First derivative for isochore 2 .......................................................................... 46
Figure 25 Second derivative for isochore 2 ..................................................................... 46
Figure 26 First derivative for isochore 3 .......................................................................... 47
Figure 27 Second derivative for isochore 3 ..................................................................... 47
Figure 28 First derivative for isochore 4 .......................................................................... 48
Figure 29 Second derivative for isochore 4 ..................................................................... 48
Figure 30 First derivative for isochore 5 .......................................................................... 49
Figure 31 Second derivative for isochore 5 ..................................................................... 49
Figure 32 First derivative for isochore 6 .......................................................................... 50
Figure 33 Second derivative for isochore 6 ..................................................................... 50
Figure 34 First derivative for isochore 7 .......................................................................... 51
Figure 35 Second derivative for isochore 7 ..................................................................... 51
Figure 36 First derivative for isochore 8 .......................................................................... 52
Figure 37 Second derivative for isochore 8 ..................................................................... 52
Figure 38 First derivative for isochore 9 .......................................................................... 53
Figure 39 Second derivative for isochore 9 ..................................................................... 53
Figure 40 First derivative for isochore 10 ........................................................................ 54
Figure 41 Second derivative for isochore 10 ................................................................... 54
Figure 42. Integrand function of (Cvr) equation for subset 1 ............................................ 57
Figure 43. Integrand function of (Cvr) equation for subset 2 ............................................ 57
Figure 44. Integrand function of (Cvr) equation for subset 3 ............................................ 58
xii
Figure 45. (Cvr) for subset 1. (220 – 300 K) ..................................................................... 59
Figure 46. (Cvr) for subset 2. (310 – 390 K) ..................................................................... 59
Figure 47. (Cvr) for subset 3. (400 – 450 K) ..................................................................... 60
Figure 48. (Cv) for subset 1. Temperatures 220 – 300 K ................................................. 63
Figure 49. (Cv) for subset 2. Temperatures 310 – 390 K ................................................. 63
Figure 50. (Cv) for subset 3. Temperatures 400 – 450 K ................................................. 64
Figure 51. Integrand function of Eq. 34 using numerical derivatives .............................. 65
Figure 52. Integrand function of Eq. 34 using analytical derivatives .............................. 66
Figure 53. Integrand function predicted. T = 280 K ........................................................ 67
Figure 54. Percentage of residual heat capacity ............................................................... 70
Figure 55. Integrand function of Eq. 34 for subset 1 using numerical derivatives ........ 118
Figure 56. Integrand function of Eq. 34 for subset 2 using numerical derivatives ........ 118
Figure 57. Integrand function of Eq. 34 for subset 3 using numerical derivatives ........ 119
xiii
LIST OF TABLES
Page
Table 1. Natural gas demand 2010 - 2018 (bcm)3 .............................................................. 2
Table 2. Relative uncertainty in (Cv) experimental values for mixtures and pure fluids
measured with adiabatic calorimeter16 ................................................................ 7
Table 3. MSD specifications ............................................................................................ 13
Table 4. Experimental measurements of density taken in the MSD of TAMU ............... 13
Table 5. Low pressure apparatus (LPI) specifications30 .................................................. 16
Table 6. High pressure apparatus (HPI) specifications6 ................................................... 16
Table 7. Experimental measurements taken in the LPI of TAMU ................................... 16
Table 8. (P-ρ-T) reference values for each isochore. EoS values from GERG-2008 ...... 23
Table 9. Thermal expansion and isothermal compressibility coefficients ....................... 25
Table 10. Volumes for LPI ............................................................................................... 26
Table 11. LPI values to calculate distortion parameters for stainless steel ...................... 27
Table 12. Parameters for stainless steel ........................................................................... 28
Table 13 Lengths for noxious volume estimation in the HPI........................................... 30
Table 14 Technical specifications for high pressure cross and valve in the HIP ............. 30
Table 15 Volumes for HPI ............................................................................................... 31
Table 16. HPI values to calculate distortion parameters for Cu-Be ................................. 33
Table 17. Parameters for Cu-Be ....................................................................................... 33
Table 18. Pseudo points in the isochoric experiment ....................................................... 38
Table 19. Outliers in the isochoric experiment ................................................................ 39
Table 20. Subset of temperatures for integral of residual heat capacity .......................... 56
xiv
Table 21. Ideal gas heat capacities at constant volume .................................................... 62
Table 22. Regression results using numerical derivatives. T = 280 K ............................. 65
Table 23. Regression results using analytical derivatives. T = 280 K ............................. 66
Table 24. Residual heat capacity at T = 280 K from numerical derivatives .................... 68
Table 25. Residual heat capacity at T = 280 K from analytical derivatives .................... 68
Table 26. Absolute heat capacity and global uncertainty at T = 280K ............................ 69
Table 27. MSD measurements and compared to values predicted by GERG 2008 ......... 79
Table 28. High pressure isochoric apparatus measurements. EoS: GERG 2008 ............. 82
Table 29. Low pressure isochoric apparatus measurements. EoS: GERG 2008 .............. 85
Table 30. Isochoric density and (P'-ρ*-T) data ................................................................ 89
Table 31. Coefficients, standard error, rms and number of points per isochore (n) for
regression analysis using Eq. 30 ....................................................................... 96
Table 32. First and second derivatives for set 1 ............................................................... 97
Table 33. First and second derivatives for set 2 ............................................................. 100
Table 34. First and second derivatives for set 3 ............................................................. 102
Table 35. Regression results of Eq. 35 for set 1 ............................................................. 106
Table 36. Regression results of Eq. 35 for set 2 ............................................................. 107
Table 37. Regression results of Eq. 35 for set 3 ............................................................. 109
Table 38. Parameters for Eq. 49 ..................................................................................... 110
Table 39. Residual and absolute heat capacity for set 1 ................................................. 111
Table 40. Residual and absolute heat capacity for set 2 ................................................. 114
Table 41. Residual and absolute heat capacity for set 3 ................................................. 116
1
1. INTRODUCTION
1.1 Economic Impact of Natural Gas
Natural gas is a mixture of several hydrocarbons, its composition depends upon
the source and it is consider as the greenest fossil fuel. Natural gases can be classify as:
“wet” when they include significant amount of hydrocarbons other than methane such as:
ethane, propane, butane and pentane; “dry” when the composition is almost pure methane,
and “sour” when it contains significant amount of hydrogen sulfide1. Natural gas is a major
energy source in the United States with consumption of 740 billion cubic meters (bcm)2
and a global demand estimated at 3,500 bcm1 in 2013, up 1.2% from 2012 levels.
The global demand for natural gases depends upon the interactions among fuel
supply and all the sectors in the custody chain of transfer, geopolitical and governmental
policies, market forces, and companies’ decisions and investments. These factors play a
role in determining the availability and competitiveness of natural gas in traditional sectors
and its expansion into new ones. The forecast for global gas demand is 3,962 bcm in 2018
as it is shown in Table 13.
Natural gas is mainly a fuel used to produce steel, glass, paper, clothing, electricity
and to heat buildings and water. It is also a raw material for fertilizers, plastics, antifreeze,
medicines and explosives2. Figure 2 exhibits the distributions for consumption of natural
gas in the U.S. in 2013.
2
Table 1. Natural gas demand 2010 - 2018 (bcm)3
Figure 1. Distribution in consumption of natural gas in U.S. 20132
3
Americas is the largest natural gas market with a growth rate of 1.5% per year in
terms of volume. Around 53% of this growth comes from the power generation sector
alone (Figure 2) in which the switch from coal to gas continues (Figure 3)3.
Figure 2. Gas demand in American market3
Figure 3. US generation from coal, gas-fired plants and renewable energies3
4
In terms of reserves of natural gas, the estimate is that about 190 trillion cubic
meters (tcm)1 of natural gas reserves exit worldwide in conventional sources4 (geological
formations that do not require specialized technologies to unlock their potential). However
the reserves from recoverable gas sources (assuming technology becomes available to
produce them) are estimated about 440 tcm1. On the other hand, estimated recoverable
unconventional resources such as Shale Gas5, trapped in reservoirs with low permeability
are around 240 tcm1. Altogether, this would last around 220 years, based upon current
rates of gas consumption1,3.
The forecast of demand suggests that study and research of the properties of natural
gas and a holistic analysis of all the factors that determine its price is crucial. When natural
gas enters to the custody transfer chain, the price is calculated from6:
3
3
$ $m kg J
day day m kg J
(1)
The first factor on the right-hand side of Eq. 1 is the volumetric flow rate
determined by flow meters (e.g. orifice meters7), the second factor is the density that is
available with high accuracy from volumetric measurements8–10. The third factor is the
heating value that is available from calorimetric measurements. The final factor is the price
of the natural gas on an energy basis. From Eq. 1 is clear that the caloric and volumetric
properties for natural gas mixtures are essential information for the gas processing
industry. Several Equations of State (EoS), such as AGA8-DC9211 and GERG-200812, can
5
predict the thermodynamic properties of natural gas. The experimental data for energies,
entropies and heat capacities are useful in developing new EoS, for empirical correlations
needed to determine the energy content of the hydrocarbon gas streams, and for use in
conjunction with other sets of experimental data, such as volumetric data (P-ρ-T), for
developing thermodynamic models13. On the other hand, knowledge of the experimental
derivatives such as P T
and 2 2P T
help in evaluating the thermodynamic
consistency of EoS and the impact of using numerical and analytical derivatives during its
development.
An important physical property of the gases is the ratio of the heat capacities at
constant pressure (CP) and heat capacities at constant volume (Cv) defined as:
P
v
Ck
C (2)
The factor k in Eq. 2 appears in the design of components and evaluation of the
overall performance of compressors14,15.
1.2. Literature Review of Calorimetric Measurements
The most popular method for determining caloric properties is via calorimetric
measurements, however those properties also can result from using thermodynamic
relationships and volumetric data. This section presents a review of the procedures and
6
current state of art for calorimetric measurements as well as the specifications and
accuracy for apparatus used to acquire volumetric data.
1.2.1 Adiabatic Calorimeter for Heat Capacities Measurements16
The National Institute of Standards and Technology (NIST) has used the adiabatic
method to measure heat capacities of liquid and gas mixtures and pure fluids. The adiabatic
calorimeter consists of a twin-bomb arrangement, which allows accurate energy
measurements. It minimizes the heat loss caused by gradients in temperature and radiation
fluxes by an automatic adjustment of the temperature of a cooper case that surrounds the
bomb. The range of temperature is from 20 to 700 K at pressures up to 35 MPa. According
to NIST the heat capacity at constant volume (Cv) is determined from:
o
v
Q QC
m T
(3)
In Eq. 3, ΔQ is the value of the energy absorbed by the fluid, ΔQ0 is the energy
difference between the empty sample bomb and the reference, ΔT is the rise in temperature
caused by the energy absorbed by a fluid of mass m contained in the bomb. Typical values
of the relative uncertainty for the measured heat capacities at constant volume appear in
Table 2.
7
Table 2. Relative uncertainty in (Cv) experimental values for mixtures and pure fluids
measured with adiabatic calorimeter16
Fluid / Mixture Phase Range Relative
Uncertainty (Cv)
Twice Distilled
Water Liquid
Temperature: 300-420 K.
Pressure: up to 20 MPa 0.48% 16
Hydrocarbon Mixture
(C3H8) and (i-C4H10) Liquid
Temperature: 203-342 K.
Pressure: up to 35 MPa 0.7% 17
Isobutane Vapor Temperature: 114-345 K.
Pressure: up to 35 MPa 0.7% 18
Propane Vapor Temperature: 85-345 K.
Pressure: up to 35 MPa 0.7% 19
Nitrogen Vapor and
Liquid
Temperature: 85-345 K.
Pressure: up to 35 MPa
2% in vapor 20
0.5% in liquid
Mixture CO2 and
C2H6
Vapor and
Liquid
Temperature: 220-340 K.
Pressure: up to 35 MPa
2% in vapor21
0.5% in liquid
1.2.2 Flow Calorimeters for Constant Pressure Heat Capacities Measurements
Ernest et al.22 constructed a flow calorimeter that accounts for the differences
between the International Practical Temperature Scale 1968 (IPTS-68) and the
Thermodynamic Temperature Scale (TTS) derived from the differences between accurate
values of the isobaric heat capacity using these corresponding scales. The design of the
calorimeter is a U-shaped tube isolated by vacuum, and a radiation shield with platinum
resistance thermometer (PRT) wires that contact the electrically heated flowing gas. The
apparatus operates continuously by recycling the gas that flows to a condenser after the
calorimeter stage followed by heating the condenser to push the liquid into a two–phase
container where the gas flow that goes to the calorimeter results at constant pressure by
evaporating the liquid. The range of operation in temperature is from -20 to 200 °C with
8
pressures up to 1.5 MPa. The experimental heat capacity at constant pressure (Cp) is
calculated from:
2 1 2 1
p
m b
PC
m T T T T
(4)
In Eq. 4 the mass flow rate m results from weighing the condensate and noting the
collection time. The Cp value results from using the difference between a main experiment
(with heating) and a blank experiment (without heating), P is the heating power in Watts.
Use of the blank experiment is convenient because it includes the effect of Joule-
Thompson cooling and changes in potential and kinetic energy, therefore improving the
accuracy of the Cp.
Measurements of specific heat capacity for Xenon in the gas phase from -20 to 100
°C and at pressures up to 0.4 MPa yield an uncertainty value for Cp of ± 0.05 %. Other
applications of flow calorimeters include the determination of enthalpies of absorption23
and enthalpy differences24.
1.2.3 Caloric Properties for Natural Gas Mixtures12
The GERG-2008 EoS12 for the thermodynamic properties of natural gases is
explicit in the Helmholtz free energy as a function of temperature, density and
composition. This EoS has a database of more than 125 000 experimental data points for
thermal and caloric properties of binary mixtures, natural gases and multicomponent
mixtures. Just 9 % of the database presents caloric properties with a relative uncertainty
9
ranging from 1 to 2% for isochoric and isobaric heat capacities and relative uncertainty
from 0.2 to 0.5% for enthalpy differences.
The caloric experimental data cover a range in temperature from 250 to 350 K at
pressures up to 20 MPa. However, compared to (P-ρ-T) data sets, measurements for
caloric properties of natural gases are scarce, especially for mixtures of nitrogen – carbon
dioxide and carbon dioxide – propane. The GERG-2008 EoS reports a percentage
deviation of ± 5 % in the values of isobaric heat capacities for ternary and quaternary
mixtures of methane, nitrogen, propane and ethane when compared to experimental
isobaric heat capacities with uncertainty in measurements of 2.5%. No values of
percentage deviation for isochoric heat capacities appear in GERG-2008.
1.3 Research Proposal and Objectives of the Study
After the analysis of the economic impact of natural gas, the importance of
experimental data for fluids and mixtures in modeling EoS is apparent. The lack of
accurate measurements for heat capacities at constant volume directly from calorimetric
measurements, requires alternatives for determining caloric properties. This research
project proposes to expand the scope of the volumetric measurements by applying
thermodynamic principles to provide accurate values of heat capacities at constant
volume.
The main objective of this work is to use (P-ρ-T) experimental data for a ternary
mixture collected in the apparatus of the Thermodynamics Research Group of Texas A&M
University for calculating the first and second derivatives of pressure with respect to
10
temperature needed for determination of the caloric properties: energies, entropies and
heat capacities. This project seeks to accomplish the following specific objectives:
1. To propose a methodology that integrates the experimental data from three
different apparatus to correct the experimental errors and provide reliable
values for the distortion coefficients for Beryllium Cooper 175 Alloy.
2. To analyze the difference between the numerical and analytical methods for
the repeatability of the results of the second derivatives of pressure with respect
to temperature at constant density 2 * 2P T
.
3. To calculate residual and absolute heat capacities from experimental (P-ρ-T)
data and to claim a value for the global uncertainty.
4. To identify a new, strategic way to develop experimental designs and data
acquisition techniques that allow building a complete thermodynamic
characterization of fluids.
The following sections present the description of the apparatus used for the
experimental characterization of gas mixtures as well as the methodology and results
obtained during the execution of this work.
11
2. EXPERIMENTAL CHARACTERIZATION OF NATURAL GAS MIXTURES
Experimental determination of thermodynamic properties for natural gas mixtures
is essential information for the gas processing industry and the development and
optimization of equations of state. The volumetric (P-ρ-T) data along with empirical
correlations can determine the energy content of hydrocarbon gas streams and their caloric
properties13. This work expands the initial scope of the (P-ρ-T) measurements by
providing a technique to obtain uncertainties for caloric properties.
The experimental data used in this work all were acquired using apparatus located
in the Thermodynamic Research Laboratory at Texas A&M University in College Station,
TX, USA. The equipment explained in this section provides accurate (P-ρ-T)
measurements of fluids over broad ranges of pressure and temperature.
2.1 Magnetic Suspension Densimeter (MSD)8,9
The MSD uses Archimedes’ principle, which states that the upward buoyancy
force exerted on a body immersed in a fluid is equal to the weight of the fluid that the body
displaces. In this apparatus the buoyant force is related to the density of the fluid measured
by weighing a sinker in vacuum and then in the presence of a fluid under pressure. The
significant feature of the apparatus is that the balance and the sinker have a magnetic
coupling (Figure 4a), thus the apparent mass measurement of the sinker occurs while it is
levitated via a suspension coupling in the pressure cell at a fixed temperature and pressure.
12
The system uses two compensation weights made from titanium (Ti) and tantalum (Ta) to
correct for nonlinearity and the effect of air buoyancy on the balance (Figure 4b).
(a) (b)
Figure 4. (a) Basic scheme for the MSD25 (b) Operating modes of the MSD25
Eq. 5 is used to determine the density of the fluid inside the pressure vessel6, in
which mv is the mass of the sinker in vacuum, ma is the apparent mass of the sinker in the
fluid, while mTi and mTa are the masses of the external weights. The volume of the sinker
(Vs) is a function of temperature and pressure. The specifications and accuracy of the MSD
appear in Table 3. The measurements using the MSD for determining the densities of pure
fluids and mixtures along with the uncertainties appear in Table 4.
13
(m ) ( )v Ti Ta a Ti Ta
s
m m m m m
V
(5)
Table 3. MSD specifications
Magnetic Suspension Densimeter
(MSD)
Uncertainty in density: 0.0005 kg/m3
Range of Density: 0- 2000 kg/m3
Material of the cell: Beryllium- Copper.
Range of Temperature: 193.15 – 523.15 K
Stability in Temperature: ± 5 mK
Pressure Range: 0-200 MPa
Uncertainty in pressure: 0.01% of full scale
Sinker (Titanium Cylinder).
Conditions measured at 293.15K
and 1 bar by Rubotherm.
Volume: 6.74083 ± 0.00034 cm3
Uncertainty in volume: ±0.05%
Mass: 30.39157 g
Analytical balance (Mettler Toledo
AT 261)
Range: 0-62 g
Uncertainty: 0.03 mg
Table 4. Experimental measurements of density taken in the MSD of TAMU
Fluid /Mixture Range of
temperature
Uncertainty in
Density
SNG 19 250 - 450 K ≤ 0.1 %
SNG (2-4)10 250 - 450 K ≤ 0.2 %
Ternary 1 (Methane, Ethane, Propane)8 300 - 400 K ≤ 0.3 %
Ethane26 298 - 450 K ≤ 0.03 %
Carbon Dioxide27 310 - 450 K ≤ 0.1 %
Methane28 300 - 450 K ≤ 0.03 %
Nitrogen29 265 - 400 K ≤ 0.05 %
14
2.2 Low and High Pressure Isochoric Apparatus6,9,30
In an isochoric experiment the amount of fluid inside the cell remains constant
while varying both temperature and pressure. Slight changes in the volume caused by
expansion and contraction of the cell material when changing temperature and pressures
along with the effect of mass transport between the cell and the pressure transducer require
correction to determine the density of the fluid. The typical arrangement of an isochoric
apparatus includes temperature and pressure controller systems, vacuum pump and
radiation shields (Figure 5). The arrangement of the radiation shields and the positions of
the thermopiles on the isochoric cell ensure rapid stability in temperature and gradients of
temperature at equilibrium of 0.005 mK6,30.
Figure 5. Schematic diagram of the isochoric experiment30
15
Figure 6 presents a general overview of the isochoric cell for the high-pressure
isochoric apparatus. The specifications of the materials, properties and accuracy of the
low and high pressure isochoric apparatus appear in Tables 5 and 6.
Figure 6. Isochoric cell cut view for the high pressure equipment6
16
Table 5. Low pressure apparatus (LPI) specifications30
Low Pressure Isochoric Apparatus (LPI)
Material of the cell, connection line and pressure transducer: Stainless Steel
Temperature: 100 – 500 K. Stability in temperature: ± 5 mK
Pressure: Up to 20 MPa. Uncertainty in pressure: 0.01% of full scale
Calibration values o
cV = 6.01*10-5 m3. o
lV = 7.32*10-9 m3. o
tV = 2.05*10-7 m3
To = 298.15 K. Po = 0.101325 MPa
Coefficients9 Stainless Steel: α = 4.86*10-5 K-1 . β = 2.53* 10-5 MPa-1
Table 6. High pressure apparatus (HPI) specifications6
High Pressure Isochoric Apparatus (HPI)
Material of the cell: Beryllium- Copper.
Material of connection line and pressure transducer: Stainless Steel
Temperature: 100 – 500 K. Stability in temperature: ± 10 mK
Pressure: Up to 200 MPa. Uncertainty in pressure: 0.01% of full scale
Calibration values = 1.05*10-5 m3. = 1.02*10-7 m3. = 1.42*10-7 m3
To = 298.15 K. Po = 0.101325 MPa
Coefficients31 Beryllium Cooper: α = 5.215*10-5 K-1 . β = 3.471* 10-5 MPa-1
Stainless Steel: α = 4.86*10-5 K-1 . β = 2.53* 10-5 MPa-1
The experimental data taken with the low pressure apparatus appear in Table 7.
The data presented in the following section represent the first reliable (P-T) data taken
with the high-pressure isochoric apparatus.
Table 7. Experimental measurements taken in the LPI of TAMU
Fluid /Mixture Range of temperature Range of pressure
SNG 19 257 - 343 K up to 20 MPa
SNG (2-4)10 250 - 350 K up to 20 MPa
Ternary 1 (Methane, Ethane, Propane)8 220 - 320 K up to 20 MPa
o
cV o
lV o
tV
17
3. HEAT CAPACITIES FROM EXPERIMENTAL (P-RHO-T) DATA
The methodology for determining the heat capacity at constant volume from
experimental data has five stages illustrated in Figure 7.
Figure 7. Methodology to determine (Cv) from experimental (P-ρ-T) data
3.1 Stage 1. Experimental Data Acquisition
This work applies the methodology from Figure 7 to a ternary mixture with a
composition similar to a residual gas sample. DCG PARTNERSHIP Inc32 prepared the
mixture gravimetrically with mole fractions of 0.95014 methane, 0.03969 ethane and
0.01017 propane with an estimated gravimetric uncertainty of ± 0.04% NIST- Traceable
by weight. Measurements of temperature, pressure and density of the sample occurred in
the MSD, LPI and HPI apparatus of the TAMU Thermodynamics Group. The set of data
18
appears in Figure 8 and Appendix A. It contains 10 isochores and 5 isotherms covering a
range of temperature from 140 to 500 K at pressures up to 200 MPa.
A statistical analysis of the properties measured over 20 minutes of stability for
each experiment indicates the set of data with the lowest standard deviation to define the
(P-T) points for each isochore and isotherm.
19
Figure 8. Experimental data of residual gas sample
0
20
40
60
80
100
120
140
160
180
200
100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500
P/
MP
a
T / K
Residual Gas Sample. CH4 = 0.95014, C2H6 = 0.03969, C3H8 = 0.01017
HPI Data
LPI Data
Phase
Boundary
MSD
Data
20
3.2 Stage 2. Isochoric Corrections
At this stage, it is necessary to consider factors that cause slight changes in density
along the experimental path. First, the volume of the cell varies with temperature and
pressure, and second, part of the volume occupied by the fluid is external to the cell and
at a different temperature. The external volume is “noxious volume” and corresponds to
the amount of gas located in the transmission line and the pressure transducer as shown in
Figure 9.
ISOCHORIC
CELL
Pressure
Transducer
Interchange of
gas moles
Transmission Line
Figure 9. Noxious volume in isochoric apparatus
21
3.2.1 Definition of Reference State
Along the “isochoric” path the values of density differ because of changes in
temperature and pressure, however the total number of moles for an isochoric line must
be constant, in the absence of leaks, in a closed system as shown in Figure 9. Section 3.2.2
describes the determination of the isochoric density based upon the total number of moles
in an isochore being constant and defined as:
c l tn n n n (6)
c c l l t tV V V V (7)
The subscripts c, l and t in Eqs. 6 and 7 and throughout this thesis, correspond to
the cell, connection line and pressure transducer respectively.
Determination of the density on the left-hand side of Eq. 7 requires a reference
state. Using the data from the MSD, it is possible to determine a (P-ρ-T) value that
corresponds to a (P-T) point on an isochore. The reference state, denoted by (P*- ρ*-T*)
for each isochore is the starting point for determining the isochoric density for each (P-T)
pair on its isochoric line.
The reference density results from using the percentage deviation in density values
from the MSD:
%deviation 100 MSD EoS
MSD
(8)
22
The Equation of State (EoS) used in Eq. 8 is GERG-2008 because it appears to
provide accurate values for densities of natural gas mixtures. Figure 10 shows the %
deviation of density for the five isotherms measured in the MSD for the residual gas
sample.
Figure 10. Percentage deviation of density values from MSD
From Figure 10, the % deviation in density has a maximum value of ± 0.05% from
20 to 200 MPa, this information along with Eq. 8 enable calculation of the reference
density (ρ*):
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0 50 100 150 200
% d
evia
tion d
ensi
ty
P / MPa
T = 300 K
T = 325 K
T = 350 K
T = 375 K
T = 400 K
23
* 100
100 %deviation
EoS
(9)
Table 8 summarizes the reference condition values for the ten isochores measured
in the high and low pressure isochoric apparatus for the residual gas sample.
Table 8. (P-ρ-T) reference values for each isochore. EoS values from GERG-2008
Isochore T* P* % deviation
ρ
Ρ EoS ρ*
(K) (MPa) (kg/m3) (kg/m3)
1 300.00 199.871 0.044 418.852 419.037
2 350.00 174.109 0.011 382.630 382.671
3 350.00 138.193 0.011 359.043 359.081
4 350.00 101.041 0.008 326.413 326.439
5 300.00 35.793 0.028 248.113 248.183
6 300.00 20.567 0.015 172.917 172.942
7 350.00 15.446 -0.001 99.564 99.563
8 350.00 7.871 0.027 49.116 49.129
9 350.00 6.383 0.022 39.374 39.383
10 350.00 4.534 0.016 27.522 27.526
3.2.2 Isochoric Density Determination
This section shows the algebraic arrangements, assumption and equations needed
to determine the isochoric density, which is the density of gas inside the isochoric cell
when at equilibrium conditions in (P-T). The isomolar characteristic of the isochoric
experiment and the reference state can correct for the noxious volume effect when
calculating the isochoric density.
24
From Eq. 6, the number of moles in the cell is
c l tn n n n (10)
Replacing the total number of moles at reference conditions:
* * *
c c l t l tn n n n n n (11)
Organizing the terms in Eq. 11 for the line and pressure transducer
* * *
c c l l t tn n n n n n (12)
Dividing Eq. 12 by the moles of the cell at reference conditions
* *
* * *1
l l t tc
c c c
n n n nn
n n n
(13)
Writing Eq. 13 in terms of density
* * * *
* * * * * *1
l l l l t t t tc c
c c c c c c
V V V VV
V V V
(14)
The assumption in the determination of isochoric density is that the main
contribution to the mass transport in the system is the expansion and contraction of the
volume of the cell when varying temperature and pressure. Furthermore, the temperature
of the connection line and pressure transducer are constant during the isochoric
experiment. Therefore, the changes in the external volume is meaningless compared to the
changes in the volume of the cell, and Eq. 14 becomes:
* *
* * * * * *1
l l l t t tc c
c c c c c c
V VV
V V V
(15)
25
Solving for the isochoric density of the cell (ρc)
*
* * *c l tc c l l t t
c c c
V V V
V V V (16)
The volume for each subsystem in Eq. 16 has the general form
* * *expV V T T P P (17)
The distortion parameters α and β in Eq. 17 are the thermal expansion and
isothermal compressibility coefficients respectively33. Table 9 contains the values of α and
β for stainless steel and Cu-Be.
1
P
V
V T
(18)
1
T
V
V P
(19)
Table 9. Thermal expansion and isothermal compressibility coefficients
Parameter Stainless Steel Beryllium Cooper
Stouffer34 Lau31 Cristancho6
a (K-1) 4.86E-05 1.85E-06 1.60E-04
β (MPa-1) 2.53E-05 3.37E-05 2.53E-05
26
Eqs. 16 and 17 are the key equations to determine the isochoric density. Because
of the different design specifications for the isochoric apparatus, the effect of the noxious
volume in the isochoric determination is analyzed separately in the following sections.
3.2.2.1 Isochoric Density for Low Pressure Isochores
The noxious volume in the LPI is mostly the volume of the pressure transducer.
The assumption is that the effect of the change in the volume of the gas when it passes
through the transmission line is negligible because the line is at the same constant
temperature as the pressure transducer, and its volume is small compared to those of the
cell and pressure transducer. Using information from Table 10, it is possible to calculate
the noxious volume for the LPI which is 0.34% of the volume of the cell. Therefore,
calculation of the densities for isochores 6 to 10 follows from:
*
* *c tc c t t
c c
V V
V V (20)
Table 10. Volumes for LPI
Low Pressure Isochoric Apparatus (LPI)
Cell
Material: Stainless Steel
Volume: 6.01·10-5 m3
Temperature of operation: 200 - 500 K
Transmission Line
Material: Stainless Steel
Volume: 7.32·10-9 m3
Temperature of operation: 350.45 K
Pressure Transducer
(43K - 101)
Material: Stainless Steel
Volume: 2.05·10-7 m3
Temperature of operation: 350.45 K
27
This research work requires validation of the parameters α and β reported in Table
9. This requires two assumptions: the total number of moles for a single isochore is
constant, and values from the MSD and GERG 2008 can provide the density in the cell
and in the pressure transducer, respectively. A system of three equations with three
unknowns (n, α and β) results from choosing three different points (P-T) on an isochore,
and determining their densities from the MSD data using the procedure outlined in Section
3.2.1. Table 11 presents the three (P-T) pairs and their density values for isochores 7, 8
and 9. By way of illustration, Eqs. 21 to 23 were used for isochore 7. Similar fashion
systems for isochores 8 and 9 provide values for n, α and β. The values appear in Table 12.
Table 11. LPI values to calculate distortion parameters for stainless steel
Isochore T P ρc ρt
(K) (MPa) (kmol/m3) (kmol/m3)
7
300.00 11.962 5.909 4.518
350.00 15.446 5.897 5.885
400.00 18.790 5.864 7.143
8
300.00 6.421 2.915 2.343
350.00 7.871 2.910 2.904
400.00 9.291 2.902 3.461
9
300.00 5.262 2.338 1.901
350.00 6.383 2.332 2.328
400.00 7.481 2.326 2.752
28
5
5
5.909*6.01 10 exp 300 350 11.962 15.446
(4.518*2.05 10 )exp 350.45 350 11.962 15.446
n
(21)
5
5
5.897*6.01 10 exp 350 350 15.446 15.446
(5.885*2.05 10 )exp 350.45 350 15.446 15.446
n
(22)
5
5
5.909*6.01 10 exp 300 350 11.962 15.446
(4.518*2.05 10 )exp 350.45 350 11.962 15.446
n
(23)
Table 12. Parameters for stainless steel
Isochore n a β
(mol) (K-1) (MPa-1)
7 0.355 4.88E-05 2.55E-05
8 0.175 4.86E-05 2.53E-05
9 0.141 4.86E-05 2.53E-05
The values for α and β reported in Table 12 validate those reported by Stouffer,
therefore Eqs. 20 and 17 along with the parameters for stainless steel reported in Table 9
are used to calculate the isochoric densities from the LPI. Figure 11 shows the % deviation
of the isochoric densities in the LPI compared to GERG 2008 using Eq. 8. The numerical
values are in Appendix B.
29
Figure 11. Per cent deviation of isochoric densities in the LPI
3.2.2.2 Isochoric Density for High Pressure Isochores
The quantification of the noxious volume and new values of α and β for the high-
pressure isochoric apparatus are new contributions from this research. To determine the
noxious volume, it is necessary to estimate the volume of the transmission line that
connects the isochoric cell to the pressure transducer. This line is stainless steel and has a
working pressure of 60,000 psi, 1/8” OD and 0.020” ID. The lengths of the each segment
of the transmission line were measured to determine the volumes reported in Table 13.
The high pressure cross and valve manufactured by HiP Co and their respective volumes
come from the technical information in Table 14. The pressure transducer, Model 430K-
101 manufactured by Paroscientific Inc., has a range of operation up to 30,000 psi and an
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
200 250 300 350 400 450 500
% d
evia
tio
n d
ensi
ty
T / K
Iso 6
Iso 7
Iso 8
Iso 9
Iso 10
30
internal volume of 0.142 cm3. Table 15 summarizes the operationing conditions and
volumes for the HPI. The initial values for determining the parameters α and β for the Cu-
Be cell are those calculated by Lau and reported in Table 9. Figure 12 illustrates the three
systems involved in the transport of mass in the HPI.
Table 13 Lengths for noxious volume estimation in the HPI
From –To Length Length Volume
(in) (cm) (cm3)
Transducer – HIP Cross 3 7.62 0.0154
HIP Cross – HIP Valve 1 2.54 0.0051
HIP Cross –Aluminum Plate 7.75 19.69 0.0399
Aluminum Plate – Isochoric Cell 5 12.7 0.0257
Total 16.75 42.55 0.0862
Table 14 Technical specifications for high pressure cross and valve in the HIP
High Pressure Cross
Model 60-24HF2
Working pressure: 60,000 psi
Connection: Tube 1/8” OD
Length (E): 1.5” = 3.81 cm
Volume: 0.0077 cm3
High Pressure Valve
Model 30-11HF2
Working pressure: 30,000 psi
Connection: Tube 1/8” OD
Length (E): 1.5” = 3.81 cm
Volume: 0.0077 cm3
31
Table 15 Volumes for HPI
High Pressure Isochoric Apparatus
Cell
Material: Beryllium Cooper 175
Volume: 1.05·10-5 m3
Temperature of operation: 200 - 500 K
Transmission Line
Material: Stainless Steel
Volume: 1.02·10-7 m3
Temperature of operation: 333.15 K
Pressure Transducer
(430K - 101)
Material: Stainless Steel
Volume: 1.42·10-7 m3
Temperature of operation: 309.15 K
From Table 15, the volume of the transmission line is approximately equal to the
volume of the pressure transducer. In addition, the temperature of operation is different
for the two systems. Consequently, both contributions must be taken into account when
calculating the noxious volume and isochoric densities from the HPI.
32
Figure 12 Layout of high pressure isochoric cell
The noxious volume for the HIP corresponds to 2.3% of the volume of the cell.
Due to the broad range in temperature and pressure in the HIP and with the aim to
determine the isochoric densities as accurate as possible, it is imperative to consider the
whole noxious volume and to determine new distortion parameters for this particular
experiment. The procedure for calculating the parameters α and β is the same as the one
Pressure Transducer
430K - 101
HIP Valve
30 K psi
HIP Cross
60 K psi
Isochoric
Cu-Be
cell
3" 1"
7" 3/4
~ 5"
Aluminum Plate
External Shield
33
explained in section 3.2.2.1 for the LPI isochores. Table 16 presents the pairs (P-T) and
density values for isochores 2, 3 and 4. The densities of the transmission line and pressure
transducer come from GERG 2008 and the densities of the cell come from MSD data.
Table 17 has the values found for n, α and β. Compared to those presented in Table 9,
parameter α differs by two orders of magnitude from the value reported by Lau, however
the value of the parameter β is fairly close. The new parameters seem to be more nearly in
agreement with those reported by Cristancho. The average in the values of α and β for
isochores 2, 3 and 4 will be used.
Table 16. HPI values to calculate distortion parameters for Cu-Be
Isochore T P ρc ρt ρl
(K) (MPa) (kmol/m3) (kmol/m3) (kmol/m3)
2
300.00 138.909 22.830 22.529 21.792
325.00 156.694 22.742 23.200 22.498
350.00 174.109 22.664 23.786 23.115
3
300.00 107.927 21.448 21.114 20.294
325.00 123.137 21.348 21.855 21.080
350.00 138.193 21.267 22.500 21.761
4 350.00 101.041 19.333 20.740 19.896
400.00 124.565 19.202 21.920 21.148
Table 17. Parameters for Cu-Be
Isochore n α β
(mol) (K-1) (MPa-1)
2 0.355 1.240E-04 3.370E-05
3 0.175 1.520E-04 3.380E-05
4 0.141 1.220E-04 3.370E-05
Average 1.327E-04 3.373E-05
34
Determining the isochoric densities from the HPI requires Eqs. 16, 17 and the
parameters for stainless steel and Cu-Be. Three different cases analyzed using the
parameters α and β for Be-Cu reported by Cristancho, Lau and the new parameters that
appear in Table 17. Figure 13 shows the % deviation of the isochoric densities from the
HPI with respect to GERG 2008 using Eq. 8. The new parameters for Be-Cu and the effect
of the noxious volume allow calculation of the gas densities from isochoric (P-T) data
with an uncertainty of ± 0.5% from 150 to 450 K at pressures up to 200 MPa. The
numerical values for the isochoric densities from the HPI using the new parameters α and
β appear in Appendix B.
Figure 13. Percentage of deviation of isochoric densities from HPI
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
100 200 300 400 500
% d
evia
tion d
ensi
ty
T / K
Cristancho
Lau
This work
35
3.2.3 True Isochores Determination
The last step in the isochoric corrections consists of transforming the experimental
pressure into values that conform to true isochores at constant densities (ρ*). To achieve
this, it is necessary to quantify the difference between the reference density (ρ*) and the
isochoric density (ρc) along the isochoric path, then multiply this difference by the change
in pressure with respect to the change in density at constant temperature. This final value
corresponds to the correction for the experimental pressure to account for the noxious
volume effect and the not truly isochore nature of the experimental data. The corrected
pressures from the isochoric experiments, denoted as (P’) are:
*
*' c
T
PP P
(24)
Because the changes in the fluid densities are small (up to 3.5%), Eq. 24 provides
sufficiently accurate pressure adjustments when using values of derivatives from an
accurate EoS such as GERG 2008. With the true isochoric set of data (P'-ρ*-T), it is
possible to determine the first and second derivatives of pressure with respect to
temperature for the 10 isochores from the LPI and the HPI. Appendix B report the values
of the derivatives used in Eq. 24 as well as the new values of (P’). Figure 14 presents the
lines that corresponds to true isochores. It is noteworthy that the change in the slopes for
the isochores from the HPI is more significant than the isochores from the LPI for which
the true isochore line lies almost on top of the experimental values.
36
0
20
40
60
80
100
120
140
160
180
200
100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500
P/
MP
a
T / K
Residual Gas Sample. CH4 = 0.95014, C2H6 = 0.03969, C3H8 = 0.01017
HIP Data
LPI Data
Phase
Boundary
MSD
Data
True
Isochore
Figure 14. True isochores diagram
37
3.3 Stage 3. Derivative Determination
After determining the true isochoric pressure (P'), the first and second derivatives
of temperature with respect to pressure at constant density come from applying the
strategy outlined in Figure 15.
Figure 15. Strategy to calculate numerical and analytical derivatives
The initial fit of pressure as function of temperature is a 5th degree polynomial for
isochores 1 to 3 and a 3rd polynomial for isochores 4 to 10.
Initial Fit
P' = f (T)
Identify and determine
pseudo points
Calculate numerical derivatives
Identify and remove outliers
Final Fit
P' = f (T)
Calculate analytical
derivatives
38
'
10
i
fit ii
P a T
(25)
Although the isochoric experiment provided measurements of P-T in 10 K
increments, some points were not measured. Thus, the next step is to establish the missing
temperatures for every isochore, the missing P-T are called “pseudo points”. This project
required three pseudo points whose values of temperature and predicted pressure using
Eq. 25 are reported in Table 18.
Table 18. Pseudo points in the isochoric experiment
Isochore T P'
(K) (MPa)
7 410 19.542
10 250 3.014
10 230 2.702
The next step is to determine the numerical derivatives for evenly spaced
temperatures. The central differences35 method provides the numerical approximation of
the first and second derivatives
02f f
f fdf
dx x
(26)
0
2
0
22
2
f f
f f fd f
dx x
(27)
39
Because the isochoric experiment contain fluctuations in temperature, pressure,
transport of gas in the noxious volume and human error, some data points are “outliers”
that are statistically inconsistent with the rest of the data36. Identification of the outliers
for this work uses subjective criteria for choosing the points that deviate from the random
or systematic tendency in the plots of the residuals in pressure (Figures 16, 18 and 20)
after fitting the data using Eq. 25 and the plots of the numerical derivatives (Figures 22 to
41) using Eqs. 26 and 27. By definition the residual of a property X is the difference
between the experimental value and the value predicted by the fit:
Residual X = Xexp – Xfit (28)
Because the central differences method for determining the numerical derivatives
always involve three points, it is possible to identify the outliers in each isochore (Table
19) that affect the calculations associated with high uncertainty in the data.
Table 19. Outliers in the isochoric experiment
T P' T P' T P'
(K) (MPa) (K) (MPa) (K) (MPa)
1 260 157.011 3 290 97.209 6 220 8.08
1 250 146.157 3 280 90.161 7 270 9.819
1 200 90.42 3 270 83.088 8 410 9.564
2 325 153.941 3 260 75.309 8 360 8.131
2 300 133.775 3 170 11.355 9 220 3.432
2 270 108.796 4 420 138.429 9 215 3.319
3 400 171.827 5 270 26.888 10 330 4.214
3 300 104.212 6 260 14.318 10 320 4.084
Isochore Isochore Isochore
40
It is valid to remove the outliers to improve the overall performance in the fit of
pressure as a function of temperature to determine the analytical derivatives. The rounded
evaluation of polynomial, exponential and rational equations used the following criteria:
1. The model must be straightforward in representing the isochoric behavior but also
determine precisely the first and second derivatives.
2. The scatter band in the plot of the residuals of pressure.
3. The coefficient values and their standard deviations.
4. The root mean square (rms) value.
2
1
1'
n
fit
i
rms P Pn
(29)
The rational equation expressed as Eq. 30 appears to be the most appropriate
equation for modeling the (P'-T) data throughout the isochoric experiment.
2
1 2 3
2
1 2 3
a a T a TP
b b T b T
(30)
The values of the coefficients, standard error, rms and number of points used per
isochore for modeling the data appear in Appendix C. The residuals in pressure calculated
using Eqs. 28 and 30 appear in Figures 17, 19 and 21. Besides the numerical and analytical
derivatives, the results of this stage indicate that the reproducibility in the values of
41
pressure is ± 0.1 MPa and ± 0.01 MPa for the measurements from the high and low
pressure isochoric apparatus respectively.
Finally, the analytical first and second derivatives of Eq. 30 appears in Eqs. 31 and
32 respectively:
2 2
1 2 3 2 3 1 2 3 2 3
22
1 2 3
2 2' a a T a T b b T b b T b T a a TP
T b b T b T
(31)
22 2 2
1 2 3 3 1 2 3 3 1 2 3
2
2 3 1 2 32
2 3 1 2 32
21 2 3 2 3
42 2
1 2 3
2 2
22 2
2'
b b T b T b a a T a T a b b T b T
b b T a a T a Tb b T b b T b T
b b T b T a a TP
T b b T b T
(32)
Because of extension of the data, the results of this section appear as plots and the
numerical values for the first and second derivatives appear in Appendix D. Figures 22 to
41 show the tendency of the first and second derivatives as functions of temperature for
the ten isochores. Furthermore, three sets of points appear per graph: the numerical
derivatives, the analytical derivatives and the values predicted by GERG 2008 EoS.
42
Figure 16 Residuals in pressure for isochores 1 to 3 using Eq. 25
Figure 17 Residuals in pressure for isochores 1 to 3 using Eq. 30
-0.5
-0.3
-0.1
0.1
0.3
100 200 300 400 500
P' -
Pfi
t
T / K
Iso 1
Iso 2
Iso 3
-0.5
-0.3
-0.1
0.1
0.3
100 200 300 400 500
P' -
Pfi
t
T / K
Iso 1
Iso 2
Iso 3
43
Figure 18 Residuals in pressure for isochores 4 and 5 using Eq. 25
Figure 19 Residuals in pressure for isochores 4 and 5 using Eq. 30
-0.02
-0.01
0
0.01
0.02
150 200 250 300 350 400 450 500
P' -
Pfi
t
T / K
Iso 4
Iso 5
-0.02
-0.01
0
0.01
0.02
150 200 250 300 350 400 450 500
P' -
Pfi
t
T / K
Iso 4
Iso 5
44
Figure 20 Residuals in pressure for isochores 6 to 10 using Eq. 25
Figure 21 Residuals in pressure for isochores 6 to 10 using Eq. 30
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
175 225 275 325 375 425 475 525
P' -
Pfi
t
T / K
Iso 6
Iso 7
Iso 8
Iso 9
Iso 10
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
175 225 275 325 375 425 475 525
P' -
Pfi
t
T / K
Iso 6
Iso 7
Iso 8
Iso 9
Iso 10
45
Figure 22 First derivative for isochore 1
Figure 23 Second derivative for isochore 1
0.9
1
1.1
1.2
1.3
1.4
125 150 175 200 225 250 275 300 325
dP
'/dT
(M
Pa/
K)
T / K
Isochore 1
GERG
Numerical
Analytical
-0.008
-0.004
0.000
0.004
0.008
125 150 175 200 225 250 275 300 325
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 1
GERG
Numerical
Analytical
46
Figure 24 First derivative for isochore 2
Figure 25 Second derivative for isochore 2
0.6
0.7
0.8
0.9
1.0
1.1
120 170 220 270 320 370 420
dP
'/dT
(M
Pa/
K)
T / K
Isochore 2
GERG
Numerical
Analytical
-0.008
-0.004
0.000
0.004
0.008
120 170 220 270 320 370 420
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 2
GERG
Numerical
Analytical
47
Figure 26 First derivative for isochore 3
Figure 27 Second derivative for isochore 3
0.5
0.6
0.7
0.8
0.9
120 160 200 240 280 320 360 400 440 480
dP
'/dT
(M
Pa/
K)
T / K
Isochore 3
GERG
Numerical
Analytical
-0.012
-0.008
-0.004
0.000
0.004
0.008
0.012
120 160 200 240 280 320 360 400 440 480
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 3
GERG
Numerical
Analytical
48
Figure 28 First derivative for isochore 4
Figure 29 Second derivative for isochore 4
0.45
0.50
0.55
0.60
300 325 350 375 400 425 450 475
dP
'/dT
(M
Pa/
K)
T / K
Isochore 4
GERG
Numerical
Analytical
-0.0006
-0.0004
-0.0002
0.0000
0.0002
300 325 350 375 400 425 450 475
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 4
GERG
Numerical
Analytical
49
Figure 30 First derivative for isochore 5
Figure 31 Second derivative for isochore 5
0.27
0.28
0.29
0.30
0.31
0.32
175 200 225 250 275 300 325
dP
'/dT
(M
Pa/
K)
T / K
Isochore 5
GERG
Numerical
Analytical
-0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
175 200 225 250 275 300 325
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 5
GERG
Numerical
Analytical
50
Figure 32 First derivative for isochore 6
Figure 33 Second derivative for isochore 6
0.150
0.152
0.154
0.156
0.158
0.160
200 220 240 260 280 300 320
dP
'/dT
(M
Pa/
K)
T / K
Isochore 6
GERG
Numerical
Analytical
-0.0005
0.0000
0.0005
0.0010
200 220 240 260 280 300 320
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 6
GERG
Numerical
Analytical
51
Figure 34 First derivative for isochore 7
Figure 35 Second derivative for isochore 7
0.060
0.065
0.070
0.075
0.080
200 250 300 350 400 450
dP
'/dT
(M
Pa/
K)
T / K
Isochore 7
GERG
Numerical
Analytical
-0.0006
-0.0003
0.0000
0.0003
0.0006
200 250 300 350 400 450
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 7
GERG
Numerical
Analytical
52
Figure 36 First derivative for isochore 8
Figure 37 Second derivative for isochore 8
0.025
0.027
0.029
0.031
0.033
200 250 300 350 400 450 500
dP
'/dT
(M
Pa/
K)
T / K
Isochore 8
GERG
Numerical
Analytical
-0.0010
-0.0005
0.0000
0.0005
0.0010
200 250 300 350 400 450 500
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 8
GERG
Numerical
Analytical
53
Figure 38 First derivative for isochore 9
Figure 39 Second derivative for isochore 9
0.015
0.020
0.025
0.030
0.035
175 225 275 325 375 425 475 525
dP
'/dT
(M
Pa/
K)
T / K
Isochore 9
GERG
Numerical
Analytical
-0.0002
-0.0001
0.0000
0.0001
0.0002
0.0003
175 225 275 325 375 425 475 525
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 9
GERG
Numerical
Analytical
54
Figure 40 First derivative for isochore 10
Figure 41 Second derivative for isochore 10
0.013
0.014
0.015
0.016
0.017
200 225 250 275 300 325 350 375 400
dP
'/dT
(M
Pa/
K)
T / K
Isochore 10
GERG
Numerical
Analytical
-0.00050
-0.00025
0.00000
0.00025
0.00050
200 225 250 275 300 325 350 375 400
d2P
'/dT
2 (M
Pa/
K2)
T / K
Isochore 10
GERG
Numerical
Analytical
55
3.4 Stage 4. Residual and Absolute Properties
A residual density function is the difference between the value of a property for a
real fluid and the value of the same property for an ideal gas at the same temperature and
density:
, , ,r igX T X T X T (33)
By transformation of thermodynamic properties (Appendix E), it is possible to
have an expression for the residual heat capacity at constant volume as function of the
second derivative of pressure with respect to temperature at constant density and the
inverse of the square of the density:
2
2 2
0
'r
v
P dC T
T
(34)
The integration indicated in Eq. 34 proceeds along isothermal paths, so it is
necessary to reorganize the isochoric data into isothermal sets to perform the integration.
Because the experimental matrix for this residual gas mixture has 10 isochores measured
at evenly spaced temperature, it would seem that every isothermal set would contain 10
pairs of points of the second derivative and the square of the molar density. Unfortunately,
the isochores do not overlap the whole range of temperature as seen in Figure 14. Table
20 presents the number of pairs and the corresponding isochores for the 24 set of isotherms
analyzed in this section.
56
Table 20. Subset of temperatures for integral of residual heat capacity
Subset 1 Subset 2 Subset 3
Temperature 220 - 300 310 - 370 380 - 390 400 - 420 430 - 450
Range (K)
Pairs of points 9 7 6 5 4
(d2P/dT2) - (ρ*2)
Isochores 1,2,3,5,6,7,8,9
and 10
2,3,4,7,8,9
and 10
2,3,4,7,8
and 9
3,4,7,8
and 9
3,4,8
and 9
Table 20 has three subsets of temperature depending upon the inclusion of the high
density isochores, i.e isochores 1 to 3. The pair of points correspond to the square of the
reference molar density (ρ*2) and the analytical second derivatives (d2P/dT2) obtained from
stage 3. The residual heat capacity calculated in this stage uses only analytical integration
of Eq. 34. The use of numerical second derivatives is usless here because of significant
scatter and no clear tendency along isotherms. Stage 5 contains an analysis of the
uncertainty in the residual heat capacity.
Figures 42 to 44 represent the integrand of Eq. 34 as a function of the reference
molar density for the three subsets of temperatures in Table 13. The integrands appear to
dictate a quadratic model represented by Eq. 35.
2
2
2
1 2 32
'P
Tc c c
(35)
57
Figure 42. Integrand function of (Cvr) equation for subset 1
Figure 43. Integrand function of (Cvr) equation for subset 2
-4.0E-06
-3.0E-06
-2.0E-06
-1.0E-06
0.0E+00
1.0E-06
0 5 10 15 20 25 30
(d2P
/dT
2)·
ρ-2
ρ* (kmol/m3)
T = 220 K
T = 230 K
T = 240 K
T = 250 K
T = 260 K
T = 270 K
T = 280 K
T = 290 K
T = 300 K
-2.0E-06
-1.5E-06
-1.0E-06
-5.0E-07
0.0E+00
0 5 10 15 20 25 30
(d2P
/dT
2)·
ρ-2
ρ* (kmol/m3)
T = 310 K
T = 320 K
T = 330 K
T = 340 K
T = 350 K
T = 360 K
T = 370 K
T = 380 K
T = 390 K
58
Figure 44. Integrand function of (Cvr) equation for subset 3
The values of the coefficients, standard error, rms, and interval of confidence of
the integrand function represented by Eq. 35 for each isotherm appear in Appendix F. The
residual heat capacity that results after analytical integration of Eqs. 35 is:
3 21 23
3 2
r
v
c cC T c
(36)
The residual values of the heat capacity calculate with Eq. 36 are compared to the
predictions from GERG 2008 EoS and appear in Figures 45 to 47 for subsets 1 to 3
respectively.
-1.E-06
-8.E-07
-6.E-07
-4.E-07
-2.E-07
0.E+00
0 5 10 15 20 25 30
(d2P
/dT
2)·
ρ-2
ρ* (kmol/m3)
T = 400 K
T = 410 K
T = 420 K
T = 430 K
T = 440 K
T = 450 K
59
Figure 45. (Cvr) for subset 1. (220 – 300 K)
Figure 46. (Cvr) for subset 2. (310 – 390 K)
0
2
4
6
8
0 5 10 15 20 25 30
Cvr
(kJ/
km
ol-
K)
ρ* (kmol/m3)
This work
GERG
0
2
4
6
8
0 5 10 15 20 25 30
Cvr
(kJ/
km
ol-
K)
ρ* (kmol/m3)
This work
GERG
60
0
2
4
6
8
0 5 10 15 20 25 30
Cvr
(kJ/
km
ol-
K)
ρ* (kmol/m3)
This work
GERG
Figure 47. (Cvr) for subset 3. (400 – 450 K)
The last step in stage 4 consists of determining the absolute value for the heat
capacity at constant volume (Cv) using residual properties:
, , ( , )r ig
v v vC T C T C T (37)
The ideal gas heat capacity at constant volume ( C
v
ig) is a function only of
temperature, for a mixture of n components with mole fraction (yi) is the weighted sum of
the individual heat capacities:
61
,i
1
nig ig
v i v
i
C y C
(38)
The Thermodynamics Research Center (TRC) has published heat capacities at
constant pressure in the ideal gas state ( C
P
ig) for methane, ethane and propane and other
components, which appear several databases and EoS such as DIPPR Project 801 and
GERG 2008 EoS. The relationship between ideal gas, heat capacities and gas constant is:
ig ig
v pC C R (39)
Complete expressions and parameters for ethane, propane and methane from the
DIPPR Project 801 appear in Appendix G. The heat capacities at constant volume
predicted by DIPPR Project 801 and GERG 2008 EoS appear in Table 21, the difference
in the values of ( C
v
ig) from both sources is about 0.05%. The values of (
C
v
ig) from GERG
2008 EoS along with the Eq. 36 provide the absolute values of (Cv) shown in Figures 48
to 50. The numerical values of the residual and absolute heat capacities determined in this
stage are reported in Appendix H and their uncertainty will be analyzed in stage 5.
62
Table 21. Ideal gas heat capacities at constant volume
T (K) Cv
ig (kJ / kmol-K)
GERG 2008 DIPPR
220 26.085 26.186
230 26.281 26.356
240 26.507 26.559
250 26.764 26.797
260 27.054 27.070
270 27.375 27.378
280 27.727 27.720
290 28.109 28.095
300 28.520 28.502
310 28.958 28.938
320 29.422 29.402
330 29.909 29.891
340 30.418 30.404
350 30.947 30.937
360 31.493 31.489
370 32.055 32.058
380 32.631 32.640
390 33.219 33.235
400 33.818 33.841
410 34.425 34.455
420 35.041 35.077
430 35.663 35.704
440 36.290 36.335
450 36.921 36.971
63
Figure 48. (Cv) for subset 1. Temperatures 220 – 300 K
Figure 49. (Cv) for subset 2. Temperatures 310 – 390 K
25
30
35
40
0 5 10 15 20 25 30
Cv
(kJ/
km
ol-
K)
ρ* (kmol/m3)
This work
GERG
25
30
35
40
0 5 10 15 20 25 30
Cv
(kJ/
km
ol-
K)
ρ* (kmol/m3)
This work
GERG
64
Figure 50. (Cv) for subset 3. Temperatures 400 – 450 K
3.5 Stage 5. Uncertainty Analysis
The last step in this methodology consists of estimating of the uncertainty in the
values of the residual and total heat capacity at constant volume. A rigorous analysis of
the numerical second derivatives would provide an estimate of the uncertainty in the
residual (Cv) from experimental (P-ρ-T) data. Even though the analysis of the data
provided values for first and second derivatives in stage 3, it was not possible to calculate
the (Cvr) from those values because their scatter (shown for every isochore in Figures 22
to 41) increases significantly when the data are available on isothermal paths.
Consequently, the numerical integration of such random data (Appendix I) is very
inaccurate for the purpose of this project and was not implemented in stage 4.
30
35
40
45
0 5 10 15 20 25 30
Cv
(kJ/
km
ol-
K)
ρ* (kmol/m3)
This work
GERG
65
However, if it is necessary to estimate the uncertainty from the data, then at least
one isotherm with smooth behavior in the integrand function of Eq. 33 using the numerical
derivative must be available. The numerical derivatives at 280 K in Figure 51 resembles
the tendency from the analytical derivatives in Figure 52.
Figure 51. Integrand function of Eq. 34 using numerical derivatives
Table 22. Regression results using numerical derivatives. T = 280 K
Coefficient Standard
Error Lower 95% Upper 95%
c1 -1.46E-08 4.72E-10 -1.86E-08 -1.06E-08
c2 3.86E-07 1.02E-08 2.80E-07 4.91E-07
c3 -2.50E-06 1.50E-07 -2.98E-06 -2.03E-06
-2.5E-06
-2.0E-06
-1.5E-06
-1.0E-06
-5.0E-07
0.0E+00
5.0E-07
0 5 10 15 20 25 30
(d2P
/dT
2)·
ρ-2
ρ* (kmol/m3)
T = 280 K - Numerical
66
The approach to determine the uncertainty in (Cv) starts with the fit of the integrand
function in Figure 51 and 52 using the same quadratic expression as in Eq. 35. The results
of the regression appear in Table 22 and 23 and Figure 53 shows the fits of the integrand
function using both numerical and analytical derivatives.
Figure 52. Integrand function of Eq. 34 using analytical derivatives
Table 23. Regression results using analytical derivatives. T = 280 K
Coefficient Standard
Error Lower 95% Upper 95%
c1 -1.30E-08 4.25E-10 -1.66E-08 -9.41E-09
c2 3.53E-07 9.14E-09 2.58E-07 4.48E-07
c3 -2.36E-06 1.35E-07 -2.78E-06 -1.93E-06
-2.5E-06
-2.0E-06
-1.5E-06
-1.0E-06
-5.0E-07
0.0E+00
5.0E-07
0 5 10 15 20 25 30
(d2P
/dT
2)·
ρ-2
ρ* (kmol/m3)
T = 280 K - Analytical
67
Figure 53. Integrand function predicted. T = 280 K
The area under the curves of Figure 47 represents the
C
v
r for both sets of data
calculated using Eq. 35 and reported in Tables 24 and 25 for the numerical and analytical
fit respectively. The difference between the values of
C
v
r is an indication of uncertainty of
the analytical method used in Stage 4.
The absolute heat capacity (Cv) results from using Eq. 36 and the (
C
v
ig ) at 280 K
from GERG 2008 EoS are reported in Table 2. The global uncertainty in the value of the
absolute heat capacity (UCv) as percentage deviation is:
%deviation *100v
analytical numerical
C
numerical
Cv CvU
Cv
(40)
-2.5E-06
-2.0E-06
-1.5E-06
-1.0E-06
-5.0E-07
0.0E+00
5.0E-07
0 5 10 15 20 25 30
(d2P
/dT
2)·
ρ-2
ρ* (kmol/m3)
Analytical
Numerical
68
Table 24. Residual heat capacity at T = 280 K from numerical derivatives
T = 280 K
Isochore P'
(MPa)
ρ*
(kmol/m3) d2P/dT2 (d2P/dT2)·ρ-2
Cvr
(kJ/kmol-K)
1 178.880 24.817 -1.25E-03 -2.02E-06 4.969
2 117.143 22.664 -7.31E-04 -1.42E-06 4.014
3 90.161 21.267 -2.35E-04 -5.19E-07 3.591
5 29.861 14.699 -2.23E-05 -1.03E-07 2.962
6 17.442 10.242 -1.99E-05 -1.90E-07 2.977
7 10.517 5.897 -2.26E-05 -6.49E-07 2.534
8 5.817 2.910 -1.35E-05 -1.59E-06 1.616
9 4.793 2.332 -7.74E-06 -1.42E-06 1.358
10 3.475 1.630 -5.54E-06 -2.08E-06 1.005
Table 25. Residual heat capacity at T = 280 K from analytical derivatives
T = 280 K
Isochore P'
(MPa)
ρ*
(kmol/m3) d2P/dT2 (d2P/dT2)·ρ-2
Cvr
(kJ/kmol-K)
1 178.880 24.817 -9.67E-04 -1.57E-06 4.474
2 117.143 22.664 -5.25E-04 -1.02E-06 3.685
3 90.161 21.267 -3.45E-04 -7.62E-07 3.342
5 29.861 14.699 -1.75E-05 -8.10E-08 2.867
6 17.442 10.242 3.48E-06 3.31E-08 2.873
7 10.517 5.897 -3.01E-05 -8.65E-07 2.420
8 5.817 2.910 -9.82E-06 -1.16E-06 1.531
9 4.793 2.332 -7.91E-06 -1.45E-06 1.285
10 3.475 1.630 -5.70E-06 -2.14E-06 0.949
Table 26 contains the absolute heat capacity (Cv) with a maximum uncertainty
(percentage deviation) of 1.5%. This estimation is an indication of the error in the
69
analytical method used to determine the residual values without including the uncertainty
of the ideal gas heat capacity.
Table 26. Absolute heat capacity and global uncertainty at T = 280K
T = 280 K
Isochore P'
(MPa)
ρ*
(kmol/m3)
Cvig
(kJ/kmol-K)
Cv (kJ/kmol-K) %
deviation Analytical Numerical
1 178.880 24.817 27.727 32.696 32.201 1.514
2 117.143 22.664 27.727 31.740 31.411 1.037
3 90.161 21.267 27.727 31.318 31.068 0.796
5 29.861 14.699 27.727 30.689 30.594 0.307
6 17.442 10.242 27.727 30.704 30.600 0.337
7 10.517 5.897 27.727 30.261 30.147 0.378
8 5.817 2.910 27.727 29.342 29.258 0.288
9 4.793 2.332 27.727 29.085 29.012 0.250
10 3.475 1.630 27.727 28.732 28.676 0.193
Finally, the values of the (Cvr) calculated in Stage 4 correspond up to 15% of the
total (Cv) as shown in Figure 54.
70
Figure 54. Percentage of residual heat capacity
0
5
10
15
20
0 5 10 15 20 25 30
(Cv
r /C
v )·
100
ρ* (kmol/m3)
71
4. CONCLUSIONS
This thesis reports highly accurate experimental (P-ρ-T) data for a ternary mixture
of methane, ethane, and propane measured with the Magnetic Suspension Densimeter
(MSD) and isochoric apparatus at Texas A&M University. The set of data contains 10
isochores and 5 isotherms covering a range of temperature from 140 to 500 K at pressures
up to 200 MPa. The relative uncertainty in pressure is 0.02 MPa for measurements up to
20 MPa, and 0.2 MPa in measurements for 20 to 200 MPa. The uncertainty in temperature
is 10 mK. From the specifications of the MSD, the relative uncertainty for density
measurements is ≤ 0.05%.
A rigorous technique allows combining the experimental data from the MSD with
the isochoric data to determine the isochoric density was proposed. This new technique,
based upon selection of a constant reference density (ρ*) for each isochore, corrects for
the effect of the noxious volume in each isochoric apparatus.
The analysis of the noxious volume for the high pressure isochoric apparatus (HPI)
is one of the main contributions of this research work. The volume of the pressure
transducer and the connection line comprise 2.3 % of the volume of the cell. New
distortion parameters for the beryllium copper alloy result from the mass balance and
algebraic manipulations of the volume expressions. The final values for the thermal
expansion and isothermal compressibility coefficients are 1.327·10-4 K-1 and 3.373·10-5
MPa respectively, which, along with the new methodology, allow determination of the
isochoric densities from the HPI within a band ± 0.5% when compared to values predicted
72
by the GERG 2008 EoS. Furthermore, the distortion parameters found in the literature
review for stainless steel were validated and allowed calculation of isochoric densities
from the low pressure isochoric apparatus (LPI) within a band of ± 0.2% for temperatures
removed from the phase loop of the ternary mixture.
The isochoric densities are not truly isochoric from the measurements. Adjusting
the data (P'-ρ*-T) such that they correspond to truly isochores is necessary when
determining the first and second derivatives of the pressure with respect to temperature at
constant density for both analytical and numerical techniques. A rational equation of
second order in both numerator and denominator fits the truly isochoric data with residuals
in pressure of ± 0.1 MPa (data from HPI) and ± 0.01 MPa (data from LPI). Moreover, this
equation could calculate derivate values having the same tendency as those predicted by
the GERG 2008 EoS.
The method of the central differences used for determining the numerical
derivatives enabled identifying those data that deviates from the tendency a result of the
experimental errors. The values of the second derivatives are about two orders of
magnitude smaller than the first derivatives and have a band of scatter of ± 0.0003 MPa/K2.
Because of the small values, the sensitivity to fluctuations of temperature and pressure of
the experimental data is noticeable. Consequently, it is important to improve the
methodology of data acquisition to ensure not only evenly space data to reduce the error
for the numerical method, but also to select accurate (P-T) points for purposes of heat
capacity determination.
73
It was possible to apply thermodynamic relationships to calculate an expression
for the residual heat capacity at constant volume that requires an isothermal integration of
the second derivative of the pressure with respect to temperature at constant density.
Twenty four isotherms covering a range from 220 to 450 K resulted from adjusting the
experimental data to true isochores. The values in the integrand function clearly reveal a
quadratic tendency when are plotted as function of the molar density. The quadratic model
agrees with the tendency in predictions from the GERG 2008 EoS and applies to all the
isotherms, even those with insufficient data. The values of the residual heat capacity result
from analytical integration of the quadratic function. Finally, the absolute heat capacities
result from using residual gas properties and ideal gas values predicted from the GERG
2008 EoS.
Although the main purpose of the (P-ρ-T) measurements is to provide accurate
data for the primary properties, it is possible to expand the initial scope and to calculate
values for the residual and absolute heat capacity at constant volume. The uncertainty
estimation requires a rigorous analysis of the numerical second derivatives because they
provide a connection to the uncertainty of the raw data. In this project, it is not possible to
calculate the residual values of the heat capacity from the numerical derivatives because
of their significant scatter on isothermal paths. Only numerical derivatives for one
isotherm (T = 280 K) were used to determine the residual heat capacity and compare them
with the values predicted from the analytical approach. The difference between both
values is an indication of the uncertainty of the (P-ρ-T) data when calculating the absolute
74
heat capacity. The maximum uncertainty in the absolute heat capacity is 1.5% with
residual values up to 15% of the absolute heat capacity at constant volume.
The value of the uncertainty in the absolute heat capacities agree with those in the
literature review for binary mixtures in the vapor phase (around 2%). Apparently, the
whole set of (P-ρ-T) data along with the numerical derivatives and heat capacities
constitute valuable input for developing equations of state such as GERG 2008, for
example, to optimize the mixing rules and parameters used in the prediction of natural gas
properties.
Finally, for future gas (P-ρ-T) measurements, the experimental design should
provide evenly spaced data in temperatures and densities to facilitate calculating
numerical derivatives and integration methods. Furthermore, a truly isochoric behavior is
achievable using a single apparatus, such as the Magnetic Suspension Densimeter (MSD),
by installing a means (e.g. hand pump) to make fine pressure adjustments and essentially
dial in the density.
75
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79
APPENDIX A. EXPERIMENTAL ISOTHERMAL AND ISOCHORIC DATA
Table 27. MSD measurements and compared to values predicted by GERG 2008
P ρMSD ρEoS 100·(ρMSD -
ρEoS)/ρMSD (MPa) (kg/m3) (kg/m3)
T = 300 K
4.999 37.306 37.287 0.051
7.503 58.646 58.62 0.045
10.002 81.394 81.357 0.046
12.505 104.869 104.819 0.048
14.999 127.822 127.793 0.022
17.508 149.468 149.437 0.02
19.995 168.813 168.788 0.015
29.984 225.81 225.738 0.032
39.987 261.352 261.289 0.024
49.935 286.23 286.088 0.05
59.928 305.314 305.16 0.051
79.952 333.809 333.646 0.049
99.947 354.975 354.803 0.049
119.913 371.914 371.736 0.048
139.983 386.188 386.007 0.047
149.8 392.43 392.245 0.047
159.85 398.411 398.221 0.048
179.73 409.211 409.025 0.045
199.494 418.862 418.676 0.044
T = 325 K
5 33.485 33.474 0.033
10 70.876 70.845 0.043
12.5 90.289 90.261 0.031
15.006 109.585 109.539 0.042
17.507 128.112 128.058 0.042
19.996 145.414 145.355 0.04
30.059 201.282 201.21 0.036
40.015 238.616 238.539 0.032
49.963 265.488 265.346 0.054
59.992 286.267 286.116 0.053
79.963 317.017 316.863 0.049
80
Table 27. Continued
P ρMSD ρEoS 100·(ρMSD - ρEoS)/ρMSD
(MPa) (kg/m3) (kg/m3)
T = 325 K
99.953 339.762 339.608 0.045
119.937 357.863 357.713 0.042
139.769 372.872 372.709 0.044
149.914 379.699 379.537 0.043
159.898 385.959 385.808 0.039
179.838 397.373 397.223 0.038
T = 350 K
1.006 5.891 5.899 -0.132
2.023 11.983 11.986 -0.024
4.998 30.470 30.465 0.016
7.495 46.652 46.639 0.027
10.021 63.431 63.418 0.021
12.526 80.209 80.209 0.000
15.025 96.805 96.806 -0.001
17.544 113.084 113.073 0.009
20.021 128.379 128.364 0.012
30.062 180.758 180.775 -0.01
40.025 218.503 218.531 -0.013
49.958 246.513 246.486 0.011
59.953 268.419 268.39 0.011
80.013 301.223 301.200 0.008
99.973 325.311 325.284 0.008
119.974 344.476 344.439 0.011
140.057 360.457 360.419 0.011
149.839 367.377 367.333 0.012
159.954 374.043 374.001 0.011
179.786 385.932 385.891 0.011
199.260 396.370 396.333 0.009
T = 375 K
1.026 5.591 5.600 -0.16
5.017 28.131 28.141 -0.034
10.019 57.578 57.575 0.005
12.508 72.366 72.364 0.003
14.970 86.847 86.843 0.005
81
Table 27. Continued
P ρMSD ρEoS 100·(ρMSD - ρEoS)/ρMSD
(MPa) (kg/m3) (kg/m3)
T = 375 K
17.509 101.434 101.431 0.003
20.011 115.297 115.300 -0.002
30.035 163.982 163.972 0.006
40.061 201.33 201.321 0.004
59.976 252.342 252.334 0.003
80.023 286.582 286.572 0.004
99.910 311.732 311.730 0.001
119.931 331.832 331.823 0.003
140.002 348.536 348.526 0.003
T = 400 K
2.054 10.539 10.543 -0.038
5.009 26.058 26.058 -0.001
10.024 52.951 52.932 0.036
12.499 66.231 66.209 0.033
15.018 79.591 79.563 0.036
17.495 92.444 92.415 0.031
20.025 105.175 105.148 0.025
30.033 150.277 150.267 0.007
40.031 186.306 186.326 -0.011
49.997 214.747 214.774 -0.012
59.979 237.723 237.756 -0.014
80.040 272.978 273.030 -0.019
100.082 299.242 299.289 -0.016
120.026 320.036 320.045 -0.003
139.961 337.257 337.263 -0.002
150.019 344.956 344.960 -0.001
82
Table 28. High pressure isochoric apparatus measurements. EoS: GERG 2008
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 1
300.00 199.871 419.037 418.851 0.044
295.00 195.624 419.424 419.219 0.049
290.00 191.147 419.817 419.500 0.076
285.00 186.633 420.213 419.785 0.102
280.00 182.090 420.612 420.080 0.126
275.00 177.422 421.015 420.336 0.161
270.00 172.833 421.418 420.656 0.181
263.06 166.346 421.984 421.084 0.213
259.99 163.152 422.245 421.124 0.265
250.00 153.629 423.076 421.771 0.308
240.00 143.91 423.923 422.439 0.350
230.00 134.009 424.789 423.143 0.387
220.00 123.882 425.676 423.863 0.426
210.00 113.535 426.588 424.619 0.462
200.00 103.186 427.523 425.555 0.460
190.00 92.171 428.512 426.281 0.521
180.00 81.107 429.544 427.194 0.547
170.00 69.828 430.641 428.217 0.563
160.00 58.336 431.833 429.385 0.567
150.00 46.673 433.170 430.769 0.554
Isochore 2
390.00 200.509 379.994 380.661 -0.176
380.00 194.055 380.655 381.150 -0.13
375.00 190.822 380.986 381.411 -0.111
370.00 187.502 381.321 381.638 -0.083
360.00 180.858 381.993 382.134 -0.037
350.00 174.109 382.671 382.63 0.011
340.00 167.260 383.356 383.133 0.058
330.00 160.301 384.049 383.639 0.107
325.00 156.694 384.400 383.84 0.146
320.00 153.255 384.749 384.164 0.152
310.00 146.102 385.458 384.702 0.196
300.00 138.909 386.174 385.298 0.227
290.00 131.404 386.906 385.778 0.292
83
Table 28. Continued
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 2
280.00 123.913 387.647 386.363 0.331
270.00 116.283 388.401 386.956 0.372
260.00 108.572 389.170 387.606 0.402
250.00 100.712 389.956 388.272 0.432
240.00 92.716 390.765 388.972 0.459
230.00 84.574 391.599 389.708 0.483
220.00 76.296 392.465 390.502 0.500
210.00 67.890 393.371 391.377 0.507
200.00 59.332 394.332 392.333 0.507
190.00 50.644 395.368 393.418 0.493
180.00 41.844 396.516 394.686 0.461
170.00 32.998 397.835 396.265 0.395
160.00 24.230 399.427 398.383 0.261
150.00 15.612 401.402 401.259 0.036
Isochore 3
450.00 193.800 352.941 354.461 -0.431
440.00 188.558 353.533 354.909 -0.389
430.00 183.199 354.131 355.327 -0.338
420.00 177.828 354.731 355.780 -0.296
410.00 172.351 355.337 356.212 -0.246
400.00 166.951 355.945 356.744 -0.224
390.00 161.350 356.560 357.195 -0.178
380.00 155.672 357.181 357.647 -0.131
370.00 149.924 357.808 358.108 -0.084
360.00 144.085 358.441 358.563 -0.034
350.00 138.193 359.081 359.043 0.011
340.00 132.236 359.727 359.543 0.051
330.00 126.204 360.382 360.058 0.090
325.00 123.137 360.713 360.304 0.113
320.00 120.066 361.046 360.567 0.133
310.00 113.859 361.719 361.104 0.170
300.00 107.927 362.392 361.970 0.116
290.00 101.518 363.089 362.533 0.153
280.00 95.010 363.801 363.112 0.189
84
Table 28. Continued
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 3
275.00 91.387 364.176 363.095 0.297
270.00 88.411 364.530 363.721 0.222
260.00 81.090 365.303 363.732 0.430
250.00 74.622 366.063 364.730 0.364
240.00 67.751 366.865 365.471 0.380
230.00 60.808 367.702 366.309 0.379
220.00 53.767 368.587 367.235 0.367
210.00 46.602 369.540 368.238 0.352
200.00 39.369 370.585 369.421 0.314
190.00 32.061 371.770 370.822 0.255
180.00 24.878 373.144 372.819 0.087
170.00 17.926 374.758 375.738 -0.262
160.00 10.359 376.726 378.293 -0.416
Isochore 4
450.00 146.879 320.809 322.241 -0.447
440.00 142.519 321.348 322.638 -0.401
430.00 138.102 321.893 323.029 -0.353
420.00 133.630 322.442 323.418 -0.303
410.00 129.127 322.995 323.826 -0.257
400.00 124.565 323.554 324.230 -0.209
390.00 119.962 324.118 324.648 -0.163
380.00 115.312 324.688 325.076 -0.120
370.00 110.611 325.264 325.515 -0.077
360.00 105.850 325.848 325.955 -0.033
350.00 101.041 326.439 326.413 0.008
340.00 96.175 327.039 326.881 0.049
330.00 91.252 327.649 327.362 0.088
320.00 86.267 328.270 327.856 0.126
310.00 81.228 328.903 328.373 0.161
85
Table 28. Continued
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 5
300.00 35.793 248.183 248.112 0.029
290.00 32.993 248.781 248.827 -0.018
280.00 30.162 249.412 249.581 -0.068
270.00 27.306 250.082 250.407 -0.130
260.00 24.432 250.798 251.358 -0.224
250.00 21.523 251.568 252.375 -0.321
240.00 18.583 252.396 253.493 -0.434
230.00 15.612 253.281 254.727 -0.571
220.00 12.599 254.209 255.982 -0.697
210.00 9.558 255.154 257.272 -0.830
200.00 6.517 256.083 258.634 -0.996
Table 29. Low pressure isochoric apparatus measurements. EoS: GERG 2008
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 6
300.00 20.567 172.942 172.917 0.015
290.00 19.023 173.065 173.066 -0.001
280.00 17.471 173.190 173.200 -0.006
270.00 15.914 173.315 173.335 -0.011
260.00 14.361 173.441 173.568 -0.073
250.00 12.795 173.569 173.698 -0.075
240.00 11.229 173.696 173.851 -0.089
230.00 9.666 173.824 174.067 -0.140
220.00 8.109 173.951 174.411 -0.264
210.00 6.534 174.079 173.006 0.616
205.00 5.806 174.141 176.458 -1.330
86
Table 29. Continued
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 7
420.00 20.120 99.115 98.833 0.284
400.00 18.790 99.241 98.989 0.255
390.00 18.128 99.305 99.103 0.203
380.00 17.461 99.369 99.208 0.162
370.00 16.791 99.434 99.315 0.119
360.00 16.116 99.498 99.413 0.086
350.00 15.446 99.563 99.565 -0.001
340.00 14.766 99.629 99.678 -0.050
330.00 14.074 99.694 99.731 -0.037
320.00 13.376 99.761 99.759 0.001
310.00 12.677 99.827 99.802 0.025
300.00 11.962 99.894 99.723 0.171
290.00 11.268 99.960 99.872 0.088
280.00 10.559 100.027 99.915 0.112
270.00 9.863 100.093 100.176 -0.082
260.00 9.136 100.160 100.112 0.048
250.00 8.413 100.228 100.170 0.057
240.00 7.691 100.295 100.376 -0.081
230.00 6.953 100.362 100.442 -0.079
220.00 6.193 100.430 100.106 0.323
Isochore 8
450.00 10.683 48.825 48.820 0.010
440.00 10.406 48.855 48.852 0.005
430.00 10.128 48.885 48.884 0.002
420.00 9.850 48.915 48.920 -0.009
410.00 9.527 48.947 48.726 0.451
400.00 9.291 48.976 48.988 -0.024
390.00 9.010 49.007 49.021 -0.030
380.00 8.727 49.037 49.048 -0.022
370.00 8.444 49.068 49.080 -0.024
360.00 8.126 49.099 48.896 0.413
350.00 7.871 49.129 49.116 0.027
340.00 7.584 49.160 49.139 0.042
330.00 7.297 49.190 49.170 0.042
87
Table 29. Continued
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 8
320.00 7.008 49.221 49.194 0.055
310.00 6.717 49.252 49.211 0.083
300.00 6.421 49.283 49.196 0.177
290.00 6.130 49.314 49.233 0.163
280.00 5.841 49.344 49.307 0.076
270.00 5.549 49.375 49.375 0.001
260.00 5.243 49.406 49.313 0.189
250.00 4.944 49.437 49.350 0.177
240.00 4.644 49.469 49.412 0.114
230.00 4.336 49.500 49.408 0.185
220.00 4.029 49.531 49.479 0.104
210.00 3.737 49.562 49.956 -0.794
Isochore 9
500.00 9.625 39.023 38.987 0.092
490.00 9.414 39.046 39.018 0.073
480.00 9.202 39.070 39.047 0.058
470.00 8.990 39.094 39.079 0.038
460.00 8.777 39.118 39.109 0.022
450.00 8.562 39.142 39.133 0.024
440.00 8.348 39.166 39.163 0.007
430.00 8.133 39.190 39.191 -0.004
420.00 7.916 39.214 39.213 0.003
410.00 7.699 39.238 39.237 0.003
400.00 7.481 39.262 39.259 0.009
390.00 7.263 39.286 39.284 0.007
380.00 7.044 39.310 39.306 0.011
370.00 6.825 39.334 39.332 0.005
360.00 6.605 39.359 39.357 0.005
350.00 6.383 39.383 39.373 0.026
340.00 6.161 39.407 39.393 0.036
330.00 5.938 39.431 39.411 0.052
320.00 5.715 39.456 39.435 0.053
310.00 5.490 39.480 39.450 0.077
300.00 5.262 39.505 39.447 0.146
88
Table 29. Continued
T P ρ ρEoS 100·(ρ - ρEoS)/ρ
(K) (MPa) (kg/m3) (kg/m3)
Isochore 9
290.00 5.038 39.529 39.485 0.112
280.00 4.812 39.553 39.516 0.094
270.00 4.585 39.578 39.551 0.067
260.00 4.353 39.602 39.550 0.133
250.00 4.122 39.627 39.574 0.132
240.00 3.891 39.651 39.622 0.074
230.00 3.663 39.676 39.745 -0.173
220.00 3.455 39.700 40.231 -1.337
215.00 3.340 39.712 40.357 -1.624
210.00 3.193 39.725 39.967 -0.610
205.00 3.088 39.737 40.333 -1.500
Isochore 10
370.00 4.822 27.493 27.447 0.168
360.00 4.682 27.509 27.506 0.011
350.00 4.534 27.526 27.521 0.017
340.00 4.384 27.543 27.526 0.061
330.01 4.214 27.560 27.394 0.600
320.00 4.084 27.576 27.541 0.126
310.00 3.939 27.593 27.591 0.008
305.15 3.868 27.601 27.612 -0.040
300.00 3.788 27.609 27.600 0.033
290.00 3.639 27.626 27.630 -0.015
280.00 3.489 27.643 27.659 -0.057
270.00 3.336 27.660 27.667 -0.027
260.00 3.182 27.677 27.673 0.014
240.00 2.874 27.710 27.708 0.007
220.00 2.561 27.744 27.740 0.016
210.01 2.402 27.761 27.750 0.042
89
APPENDIX B. ISOCHORIC DENSITY AND TRULY ISOCHORIC DATA
Table 30. Isochoric density and (P'-ρ*-T) data
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 1. ρ* = 419.037 kg/m3
300.00 199.871 419.037 0.044 2.159 199.871
295.00 195.624 419.424 0.049 2.130 194.801
290.00 191.147 419.817 0.076 2.100 189.510
285.00 186.633 420.213 0.102 2.070 184.199
280.00 182.090 420.612 0.126 2.039 178.880
275.00 177.422 421.015 0.161 2.008 173.449
270.00 172.833 421.418 0.181 1.977 168.125
263.06 166.346 421.984 0.213 1.934 160.647
259.99 163.152 422.245 0.265 1.914 157.011
250.00 153.629 423.076 0.309 1.850 146.157
240.00 143.910 423.923 0.350 1.784 135.191
230.00 134.009 424.789 0.388 1.717 124.133
220.00 123.882 425.676 0.426 1.648 112.942
210.00 113.535 426.588 0.462 1.577 101.626
200.00 103.186 427.523 0.460 1.504 90.420
190.00 92.171 428.512 0.521 1.429 78.628
180.00 81.107 429.544 0.547 1.352 66.902
170.00 69.828 430.641 0.563 1.272 55.070
160.00 58.336 431.833 0.567 1.189 43.128
150.00 46.673 433.170 0.554 1.102 31.100
Isochore 2. ρ* = 382.671 kg/m3
390.00 200.509 379.994 -0.176 1.899 205.593
380.00 194.055 380.655 -0.130 1.854 197.792
375.00 190.822 380.986 -0.111 1.831 193.907
370.00 187.502 381.321 -0.083 1.808 189.943
360.00 180.858 381.993 -0.037 1.761 182.052
350.00 174.109 382.671 0.011 1.713 174.109
340.00 167.260 383.356 0.058 1.665 166.119
330.00 160.301 384.049 0.107 1.617 158.073
325.00 156.694 384.400 0.146 1.592 153.941
90
Table 30. Continued.
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 2. ρ* = 382.671 kg/m3
320.00 153.255 384.749 0.152 1.567 149.998
310.00 146.102 385.458 0.196 1.517 141.875
300.00 138.909 386.174 0.227 1.466 133.775
290.00 131.404 386.906 0.292 1.414 125.417
280.00 123.913 387.647 0.331 1.361 117.143
270.00 116.283 388.401 0.372 1.307 108.796
260.00 108.572 389.170 0.402 1.252 100.439
250.00 100.712 389.956 0.432 1.195 92.003
240.00 92.716 390.765 0.459 1.138 83.506
230.00 84.574 391.599 0.483 1.079 74.941
220.00 76.296 392.465 0.500 1.019 66.321
210.00 67.890 393.371 0.507 0.957 57.655
200.00 59.332 394.332 0.507 0.893 48.923
190.00 50.644 395.368 0.493 0.827 40.148
180.00 41.844 396.516 0.461 0.758 31.346
170.00 32.998 397.835 0.395 0.687 22.579
160.00 24.230 399.427 0.261 0.613 13.967
150.00 15.612 401.402 0.036 0.533 5.620
Isochore 3. ρ* = 359.081 kg/m3
450.00 193.800 352.941 -0.431 1.752 204.556
440.00 188.558 353.533 -0.389 1.713 198.062
430.00 183.199 354.131 -0.338 1.674 191.487
420.00 177.828 354.731 -0.296 1.635 184.939
410.00 172.351 355.337 -0.246 1.595 178.322
400.00 166.951 355.945 -0.224 1.555 171.827
390.00 161.350 356.560 -0.178 1.514 165.166
380.00 155.672 357.181 -0.131 1.473 158.470
370.00 149.924 357.808 -0.084 1.431 151.746
360.00 144.085 358.441 -0.034 1.389 144.973
350.00 138.193 359.081 0.011 1.346 138.193
340.00 132.236 359.727 0.051 1.302 131.394
330.00 126.204 360.382 0.090 1.258 124.567
325.00 123.137 360.713 0.113 1.236 121.120
91
Table 30. Continued
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 3. ρ* = 359.081 kg/m3
320.00 120.066 361.046 0.133 1.214 117.681
310.00 113.859 361.719 0.170 1.168 110.777
300.00 107.927 362.392 0.116 1.122 104.212
290.00 101.518 363.089 0.153 1.075 97.209
280.00 95.010 363.801 0.189 1.027 90.161
275.00 91.387 364.176 0.297 1.003 86.277
270.00 88.421 364.529 0.222 0.979 83.088
260.00 81.090 365.303 0.430 0.929 75.309
250.00 74.622 366.063 0.364 0.878 68.488
240.00 67.751 366.865 0.380 0.827 61.316
230.00 60.808 367.702 0.379 0.774 54.138
220.00 53.767 368.587 0.367 0.719 46.929
210.00 46.602 369.540 0.352 0.664 39.662
200.00 39.369 370.585 0.314 0.606 32.398
190.00 32.061 371.770 0.255 0.546 25.128
180.00 24.878 373.144 0.087 0.484 18.067
170.00 17.926 374.758 -0.262 0.419 11.355
160.00 10.359 376.726 -0.416 0.350 4.192
Isochore 4. ρ* = 326.439 kg/m3
450.00 146.879 320.809 -0.447 1.301 154.205
440.00 142.519 321.348 -0.401 1.268 148.974
430.00 138.102 321.893 -0.353 1.234 143.714
420.00 133.630 322.442 -0.303 1.200 138.429
410.00 129.127 322.995 -0.257 1.166 133.144
400.00 124.565 323.554 -0.209 1.132 127.830
390.00 119.962 324.118 -0.163 1.097 122.508
380.00 115.312 324.688 -0.120 1.061 117.171
370.00 110.611 325.264 -0.077 1.025 111.816
360.00 105.850 325.848 -0.033 0.989 106.435
350.00 101.041 326.439 0.008 0.953 101.041
340.00 96.175 327.039 0.049 0.916 95.626
330.00 91.252 327.649 0.088 0.878 90.190
320.00 86.267 328.270 0.126 0.840 84.729
310.00 81.228 328.903 0.161 0.801 79.253
92
Table 30. Continued
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 5. ρ* = 248.183 kg/m3
300.00 35.793 248.183 0.028 0.294 35.793
290.00 32.993 248.781 -0.019 0.270 32.832
280.00 30.162 249.412 -0.068 0.245 29.861
270.00 27.306 250.082 -0.131 0.220 26.888
260.00 24.432 250.798 -0.223 0.195 23.921
250.00 21.523 251.568 -0.320 0.170 20.947
240.00 18.583 252.396 -0.436 0.144 17.975
230.00 15.612 253.281 -0.570 0.119 15.007
220.00 12.599 254.209 -0.698 0.093 12.041
210.00 9.558 255.155 -0.828 0.066 9.094
200.00 6.517 256.083 -0.994 0.040 6.204
Isochore 6. ρ* = 172.942 kg/m3
300.00 20.567 172.942 0.015 0.141 20.567
290.00 19.023 173.065 -0.001 0.126 19.007
280.00 17.471 173.189 -0.006 0.112 17.444
270.00 15.914 173.315 -0.011 0.097 15.878
260.00 14.361 173.441 -0.074 0.083 14.32
250.00 12.795 173.568 -0.075 0.069 12.752
240.00 11.229 173.695 -0.09 0.054 11.188
230.00 9.666 173.822 -0.141 0.041 9.631
220.00 8.109 173.949 -0.265 0.027 8.081
210.00 6.534 174.077 0.615 0.014 6.517
205.00 5.806 174.138 -1.332 0.009 5.796
Isochore 7. ρ* = 99.563 kg/m3
420.00 20.120 99.113 0.282 0.217 20.218
400.00 18.790 99.241 0.254 0.199 18.854
390.00 18.128 99.305 0.203 0.190 18.177
380.00 17.461 99.369 0.161 0.181 17.496
370.00 16.791 99.433 0.119 0.171 16.813
360.00 16.116 99.498 0.086 0.162 16.127
350.00 15.446 99.563 -0.001 0.153 15.446
340.00 14.766 99.629 -0.050 0.144 14.757
330.00 14.074 99.695 -0.037 0.134 14.057
93
Table 30. Continued
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 7. ρ* = 99.563 kg/m3
320.00 13.376 99.761 0.001 0.125 13.351
310.00 12.677 99.827 0.025 0.116 12.647
300.00 11.962 99.894 0.171 0.106 11.927
290.00 11.268 99.960 0.088 0.097 11.230
280.00 10.559 100.027 0.112 0.088 10.519
270.00 9.863 100.093 -0.082 0.078 9.821
260.00 9.136 100.160 0.048 0.069 9.094
250.00 8.413 100.227 0.057 0.059 8.374
240.00 7.691 100.294 -0.082 0.050 7.655
230.00 6.953 100.362 -0.080 0.041 6.920
220.00 6.193 100.429 0.322 0.031 6.166
Isochore 8. ρ* = 49.129 kg/m3
450.00 10.683 48.824 0.008 0.220 10.750
440.00 10.406 48.854 0.003 0.213 10.465
430.00 10.128 48.884 0.001 0.206 10.179
420.00 9.850 48.915 -0.010 0.199 9.893
410.00 9.527 48.946 0.450 0.193 9.562
400.00 9.291 48.976 -0.025 0.186 9.319
390.00 9.010 49.006 -0.030 0.179 9.032
380.00 8.727 49.037 -0.022 0.172 8.743
370.00 8.444 49.068 -0.025 0.165 8.454
360.00 8.126 49.099 0.413 0.158 8.131
350.00 7.871 49.129 0.027 0.152 7.871
340.00 7.584 49.160 0.042 0.145 7.580
330.00 7.297 49.190 0.042 0.138 7.288
320.00 7.008 49.221 0.056 0.131 6.996
310.00 6.717 49.252 0.083 0.124 6.702
300.00 6.421 49.283 0.177 0.117 6.403
290.00 6.130 49.314 0.164 0.110 6.110
280.00 5.841 49.344 0.076 0.103 5.819
270.00 5.549 49.375 0.001 0.095 5.526
260.00 5.243 49.406 0.189 0.088 5.219
250.00 4.944 49.437 0.177 0.081 4.919
94
Table 30. Continued
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 8. ρ* = 49.129 kg/m3
240.00 4.644 49.468 0.114 0.074 4.619
230.00 4.336 49.499 0.184 0.066 4.311
220.00 4.029 49.531 0.104 0.059 4.005
210.00 3.737 49.561 -0.795 0.051 3.715
Isochore 9. ρ* = 39.383 kg/m3
500.00 9.625 39.021 0.088 0.250 9.715
490.00 9.414 39.045 0.069 0.244 9.497
480.00 9.202 39.069 0.055 0.237 9.277
470.00 8.990 39.093 0.035 0.231 9.057
460.00 8.777 39.117 0.020 0.225 8.837
450.00 8.562 39.141 0.022 0.218 8.615
440.00 8.348 39.165 0.005 0.212 8.394
430.00 8.133 39.189 -0.006 0.206 8.172
420.00 7.916 39.213 0.001 0.199 7.949
410.00 7.699 39.238 0.002 0.193 7.727
400.00 7.481 39.262 0.008 0.186 7.504
390.00 7.263 39.286 0.006 0.180 7.281
380.00 7.044 39.310 0.010 0.173 7.057
370.00 6.825 39.334 0.005 0.167 6.833
360.00 6.605 39.359 0.004 0.161 6.608
350.00 6.383 39.383 0.026 0.154 6.383
340.00 6.161 39.407 0.036 0.148 6.158
330.00 5.938 39.432 0.052 0.141 5.932
320.00 5.715 39.456 0.053 0.134 5.705
310.00 5.490 39.480 0.077 0.128 5.477
300.00 5.262 39.505 0.146 0.121 5.248
290.00 5.038 39.529 0.112 0.115 5.021
280.00 4.812 39.553 0.094 0.108 4.794
270.00 4.585 39.578 0.067 0.101 4.565
260.00 4.353 39.602 0.132 0.094 4.332
250.00 4.122 39.627 0.132 0.088 4.100
240.00 3.891 39.651 0.074 0.081 3.869
230.00 3.663 39.676 -0.174 0.074 3.642
95
Table 30. Continued
T P ρc 100·(ρc-ρEoS)/ρc
(dP/dρ*)T P'
(K) (MPa) (kg/m3) (MPa·m3/kg) (MPa)
Isochore 9. ρ* = 39.383 kg/m3
220.00 3.455 39.700 -1.338 0.067 3.433
215.00 3.340 39.712 -1.624 0.063 3.320
210.00 3.193 39.725 -0.611 0.059 3.173
205.00 3.088 39.737 -1.501 0.056 3.069
Isochore 10. ρ* = 27.526 kg/m3
370.00 4.822 27.493 0.168 0.170 4.828
360.00 4.682 27.509 0.010 0.164 4.684
350.00 4.534 27.526 0.017 0.158 4.534
340.00 4.384 27.543 0.061 0.152 4.381
330.01 4.214 27.560 0.600 0.146 4.209
320.00 4.084 27.576 0.126 0.140 4.077
310.00 3.939 27.593 0.008 0.134 3.930
305.15 3.868 27.601 -0.040 0.131 3.858
300.00 3.788 27.609 0.033 0.128 3.778
290.00 3.639 27.626 -0.015 0.122 3.627
280.00 3.489 27.643 -0.057 0.116 3.476
270.00 3.336 27.660 -0.027 0.109 3.321
260.00 3.182 27.677 0.014 0.103 3.167
240.00 2.874 27.710 0.006 0.091 2.857
220.00 2.561 27.744 0.015 0.078 2.544
210.01 2.402 27.761 0.041 0.071 2.386
96
APPENDIX C. REGRESSION RESULTS OF FIT P VS T
Table 31. Coefficients, standard error, rms and number of points per isochore (n) for regression analysis using Eq. 30
Isochore Parameter a1 a2 a3 b1 b2 b3 rms n
1 Coefficient 10.009 -1.75E-01 7.57E-04 -6.15E-02 5.45E-04 2.96E-07
4.71E-03 17 Standard error 1.73E-03 9.09E-06 4.27E-08 7.35E-06 1.60E-07 1.28E-11
2 Coefficient 9.052 -1.27E-01 4.46E-04 -6.49E-02 4.43E-04 1.65E-07
1.23E-03 24 Standard error 5.98E-05 3.63E-07 2.12E-09 2.70E-06 3.49E-08 1.55E-12
3 Coefficient 11.148 -1.50E-01 5.03E-04 -8.92E-02 6.11E-04 1.85E-07
1.53E-03 25 Standard error 6.50E-04 3.34E-06 1.42E-08 4.82E-06 8.98E-08 3.16E-12
4 Coefficient 2.332 -5.53E+00 3.21E-02 -1.02E+00 5.72E-02 6.50E-06
2.32E-05 15 Standard error 4.83E-02 1.30E-04 3.40E-07 8.69E-07 1.09E-06 5.51E-12
5 Coefficient 15.119 -1.65E-01 4.52E-04 -2.78E-01 1.48E-03 6.62E-08
5.31E-06 11 Standard error 7.33E-05 3.42E-07 1.56E-09 3.41E-05 2.75E-07 1.53E-13
6 Coefficient 10.718 -1.16E-01 3.13E-04 -4.08E-01 2.01E-03 -1.85E-08
6.27E-06 9 Standard error 1.87E-05 9.11E-08 4.43E-10 1.54E-04 1.07E-06 2.58E-14
7 Coefficient 20.197 -2.40E-01 6.75E-04 -1.89E+00 8.23E-03 1.92E-06
4.27E-05 18 Standard error 1.12E-04 5.10E-07 2.31E-09 5.26E-04 3.75E-06 7.57E-11
8 Coefficient -17.486 1.71E-01 2.06E-04 5.61E+00 8.65E-03 -5.34E-07
1.38E-05 23 Standard error 1.12E-02 3.46E-05 9.77E-08 6.53E-04 5.50E-07 6.78E-13
9 Coefficient 19.567 -3.55E-01 1.26E-03 -1.08E+01 5.11E-02 5.44E-06
1.52E-06 29 Standard error 1.87E-04 9.03E-07 4.32E-09 5.82E-04 5.71E-06 2.74E-11
10 Coefficient 13.357 -3.23E-01 1.57E-03 -1.41E+01 9.37E-02 1.45E-05
2.88E-06 14 Standard error 9.84E-03 3.81E-05 1.38E-07 2.60E-03 3.76E-05 2.84E-10
97
APPENDIX D. FIRST AND SECOND DERIVATIVES
Table 32. First and second derivatives for set 1
Isochore
P' ρ* Analytical Derivatives Numerical Derivatives
(MPa) (kmol/m3) (dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 220 K
1 112.942 24.817 1.1291 -1.07E-03 1.1253 -1.25E-03
2 66.321 22.664 0.8634 -5.05E-04 0.8643 -4.44E-04
3 46.929 21.267 0.7227 -2.09E-04 0.7238 -5.76E-04
5 12.041 14.699 0.2958 1.51E-04 0.2956 1.95E-04
6 8.08 10.242 0.1558 5.97E-05 0.1556 -1.50E-04
8 4.004 2.91 0.0304 -1.28E-05 0.0298 1.54E-04
9 3.432 2.332 0.0233 -1.79E-05 0.0322 3.79E-03
10 2.543 1.63 0.0158 -9.80E-06 0.0159 9.10E-06
T = 230 K
1 124.133 24.817 1.1186 -1.05E-03 1.1128 -1.34E-03
2 74.941 22.664 0.8582 -5.19E-04 0.8593 -5.59E-04
3 54.138 21.267 0.7203 -2.63E-04 0.7194 -3.17E-04
5 15.007 14.699 0.2968 5.60E-05 0.2967 2.44E-05
6 9.629 10.242 0.1561 1.67E-05 0.1553 8.69E-05
7 6.918 5.897 0.073 -7.65E-05 -0.0002 -1.88E-04
8 4.31 2.91 0.0303 -1.22E-05 0.0307 1.61E-05
9 3.641 2.332 0.0232 -7.08E-06 0.0218 1.93E-04
10 2.702 1.63 0.0157 -9.80E-06 0.0157 -4.13E-05
T = 240 K
1 135.191 24.817 1.1082 -1.03E-03 1.101 -9.18E-04
2 83.506 22.664 0.853 -5.25E-04 0.8531 -6.77E-04
3 61.316 21.267 0.7175 -2.95E-04 0.7175 -5.20E-05
5 17.975 14.699 0.2971 1.85E-05 0.297 4.04E-05
6 11.186 10.242 0.1562 8.21E-06 0.1561 6.71E-05
7 7.653 5.897 0.0725 -3.72E-05 0.0727 -1.56E-04
8 4.618 2.91 0.0301 -1.17E-05 0.0304 -7.31E-05
9 3.868 2.332 0.0231 -5.49E-06 0.0229 3.29E-05
10 2.856 1.63 0.0156 -8.25E-06 0.0156 2.62E-05
98
Table 32. Continued
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 250 K
1 146.16 24.82 1.10 -1.01E-03 1.0911 -1.12E-03
2 92.00 22.66 0.85 -5.28E-04 0.8466 -6.20E-04
3 68.49 21.27 0.71 -3.16E-04 0.6996 -3.52E-03
5 20.95 14.70 0.30 7.44E-07 0.2973 1.15E-05
6 12.75 10.24 0.16 5.47E-06 0.1566 3.44E-05
7 8.37 5.90 0.07 -3.24E-05 0.072 1.44E-05
8 4.92 2.91 0.03 -1.12E-05 0.03 -5.88E-06
9 4.10 2.33 0.02 -5.04E-06 0.0231 7.73E-06
10 3.01 1.63 0.02 -7.24E-06 0.0155 -5.05E-05
T = 260 K
1 157.011 24.817 1.0879 -9.97E-04 1.0991 -1.12E-03
2 100.439 22.664 0.8319 -5.28E-04 0.8396 -6.20E-04
3 75.309 21.267 0.7112 -3.30E-04 0.73 9.60E-03
5 23.921 14.699 0.2972 -8.67E-06 0.297 -6.70E-05
6 14.318 10.242 0.1563 4.33E-06 0.1563 -9.16E-05
7 9.092 5.897 0.0719 -3.11E-05 0.0724 6.61E-05
8 5.218 2.91 0.0299 -1.07E-05 0.0303 7.23E-05
9 4.331 2.332 0.023 -4.86E-06 0.0233 1.57E-05
10 3.166 1.63 0.0154 -6.55E-06 0.0153 1.99E-05
T = 270 K
1 168.125 24.817 1.078 -9.82E-04 1.0925 -3.58E-03
2 108.796 22.664 0.8372 -5.27E-04 0.8352 -1.03E-04
3 83.088 21.267 0.7078 -3.39E-04 0.7426 -7.08E-03
5 26.888 14.699 0.2971 -1.41E-05 0.297 6.46E-05
6 15.876 10.242 0.1564 3.78E-06 0.1562 7.71E-05
7 9.819 5.897 0.0716 -3.05E-05 0.0712 -2.95E-04
8 5.525 2.91 0.0298 -1.03E-05 0.03 -1.41E-04
9 4.564 2.332 0.023 -4.77E-06 0.0231 -4.65E-05
10 3.32 1.63 0.0154 -6.06E-06 0.0154 2.58E-06
99
Table 32. Continued
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 280 K
1 178.88 24.817 1.0683 -9.67E-04 1.0692 -1.25E-03
2 117.143 22.664 0.8319 -5.25E-04 0.831 -7.31E-04
3 90.161 21.267 0.7044 -3.45E-04 0.706 -2.35E-04
5 29.861 14.699 0.2969 -1.75E-05 0.2972 -2.23E-05
6 17.442 10.242 0.1564 3.48E-06 0.1565 -1.99E-05
7 10.517 5.897 0.0713 -3.01E-05 0.0704 -2.26E-05
8 5.817 2.91 0.0297 -9.82E-06 0.0292 -1.35E-05
9 4.793 2.332 0.0229 -7.91E-06 0.0228 -7.74E-06
10 3.475 1.63 0.0153 -5.70E-06 0.0153 -5.54E-06
T = 290 K
1 189.51 24.817 1.0587 -9.53E-04 1.0496 -2.69E-03
2 125.417 22.664 0.8267 -5.22E-04 0.8316 8.49E-04
3 97.209 21.267 0.701 -3.48E-04 0.7026 -4.57E-04
5 32.832 14.699 0.2967 -1.97E-05 0.2966 -9.67E-05
6 19.007 10.242 0.1564 3.30E-06 0.1563 -3.63E-05
7 11.228 5.897 0.071 -2.98E-05 0.0704 -1.40E-04
8 6.109 2.91 0.0296 -9.40E-06 0.0292 1.51E-05
9 5.02 2.332 0.0229 -4.67E-06 0.0227 -4.02E-06
10 3.626 1.63 0.0153 -5.43E-06 0.0151 -7.35E-06
T = 300 K
1 199.871 24.817 1.0492 -9.40E-04 N/A N/A
2 133.775 22.664 0.8215 -5.19E-04 0.8229 -2.59E-03
3 104.212 21.267 0.6975 -3.50E-04 0.6784 -4.38E-03
5 35.793 14.699 0.2965 -2.12E-05 N/A N/A
6 20.567 10.242 0.1565 3.19E-06 N/A N/A
7 11.925 5.897 0.0707 -2.95E-05 0.0709 2.30E-04
8 6.402 2.91 0.0295 -9.00E-06 0.0296 6.65E-05
9 5.247 2.332 0.0228 -4.64E-06 0.0228 3.15E-05
10 3.777 1.63 0.0152 -5.22E-06 0.0152 1.45E-05
100
Table 33. First and second derivatives for set 2
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 310 K
2 141.875 22.664 0.8163 -5.15E-04 0.811 2.36E-04
3 110.777 21.267 0.694 -3.51E-04 0.673 3.39E-03
4 79.253 19.333 0.5491 -2.36E-04 N/A N/A
7 12.645 5.897 0.0704 -2.93E-05 0.071 -1.55E-04
8 6.701 2.91 0.0294 -8.62E-06 0.03 -6.11E-05
9 5.477 2.332 0.0227 -4.62E-06 0.023 -2.36E-05
10 3.929 1.63 0.0151 -5.05E-06 0.015 -5.40E-05
T = 320 K
2 149.998 22.664 0.8112 -5.11E-04 0.8099 -4.84E-04
3 117.681 21.267 0.6905 -3.51E-04 0.6895 -1.87E-04
4 84.729 19.333 0.5468 -2.24E-04 0.5468 -1.43E-04
7 13.35 5.897 0.0701 -2.91E-05 0.0705 1.42E-05
8 6.995 2.91 0.0294 -8.25E-06 0.0293 -7.92E-06
9 5.704 2.332 0.0227 -4.60E-06 0.0227 -6.73E-06
10 4.076 1.63 0.0151 -4.92E-06 0.014 -1.47E-04
T = 330 K
2 158.073 22.664 0.8061 -5.07E-04 0.806 -2.91E-04
3 124.567 21.267 0.6869 -3.51E-04 0.6856 -5.77E-04
4 90.19 19.333 0.5446 -2.14E-04 0.5449 -2.49E-04
7 14.056 5.897 0.0698 -2.89E-05 0.0703 -5.51E-05
8 7.288 2.91 0.0293 -7.90E-06 0.0292 -5.54E-06
9 5.931 2.332 0.0227 -4.58E-06 0.0226 -7.68E-06
10 4.208 1.63 0.015 -4.81E-06 0.0152 4.05E-04
T = 340 K
2 166.119 22.664 0.801 -5.03E-04 0.8018 -5.64E-04
3 131.394 21.267 0.6834 -3.50E-04 0.6813 -2.93E-04
4 95.626 19.333 0.5425 -2.04E-04 0.5426 -2.10E-04
7 14.756 5.897 0.0695 -2.87E-05 0.0695 -1.10E-04
8 7.58 2.91 0.0292 -7.56E-06 0.0292 -9.99E-06
9 6.157 2.332 0.0226 -4.56E-06 0.0226 -1.98E-06
10 4.381 1.63 0.015 -4.72E-06 0.0163 -2.03E-04
101
Table 33. Continued
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 350 K
2 174.109 22.664 0.796 -4.99E-04 0.7967 -5.64E-04
3 138.193 21.267 0.6799 -3.49E-04 0.679 -1.76E-04
4 101.041 19.333 0.5405 -1.96E-04 0.5405 -2.11E-04
7 15.446 5.897 0.0692 -2.86E-05 0.0686 -8.39E-05
8 7.871 2.91 0.0291 -7.23E-06 0.0276 -3.12E-04
9 6.383 2.332 0.0226 -4.55E-06 0.0226 -5.37E-06
10 4.534 1.63 0.015 -4.64E-06 0.0152 -1.28E-05
T = 360 K
2 182.052 22.664 0.791 -4.95E-04 0.7917 -5.32E-04
3 144.973 21.267 0.6765 -3.47E-04 0.6777 -8.13E-05
4 106.435 19.333 0.5386 -1.89E-04 0.5387 -1.32E-04
7 16.127 5.897 0.0689 -2.84E-05 0.0684 6.28E-05
8 8.131 2.91 0.0291 -6.92E-06 0.0292 6.41E-04
9 6.609 2.332 0.0225 -4.53E-06 0.0225 -6.76E-06
10 4.685 1.63 0.0149 -4.58E-06 0.0147 -7.52E-05
T = 370 K
2 189.943 22.664 0.7837 -4.90E-04 0.787 -4.05E-04
3 151.746 21.267 0.673 -3.46E-04 0.6748 -4.85E-04
4 111.816 19.333 0.5367 -1.82E-04 0.5368 -2.56E-04
7 16.815 5.897 0.0686 -2.82E-05 0.0685 -4.17E-05
8 8.455 2.91 0.029 -6.62E-06 0.0307 -3.50E-04
9 6.833 2.332 0.0225 -4.52E-06 0.0224 -4.26E-06
10 4.828 1.63 0.0149 -4.52E-06 N/A N/A
T = 380 K
2 197.792 22.664 0.7812 -4.86E-04 0.7825 -4.95E-04
3 158.47 21.267 0.6696 -3.44E-04 0.671 -2.83E-04
4 117.171 19.333 0.535 -1.77E-04 0.5346 -1.82E-04
7 17.498 5.897 0.0684 -2.80E-05 0.0685 -1.44E-05
8 8.744 2.91 0.0289 -6.33E-06 0.0289 2.46E-07
9 7.058 2.332 0.0224 -4.50E-06 0.0224 -2.44E-06
102
Table 33. Continued
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 390 K
2 205.593 22.664 0.776 -4.82E-04 N/A N/A
3 165.166 21.267 0.666 -3.42E-04 0.668 -3.48E-04
4 122.508 19.333 0.533 -1.71E-04 0.533 -1.45E-04
7 18.18 5.897 0.068 -2.78E-05 0.068 -4.41E-05
8 9.033 2.91 0.029 -6.06E-06 0.029 -1.45E-05
9 7.282 2.332 0.022 -4.49E-06 0.022 -3.98E-06
Table 34. First and second derivatives for set 3
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 400 K
3 171.827 21.267 0.6627 -3.40E-04 0.6578 -1.66E-03
4 127.83 19.333 0.5315 -1.66E-04 0.5318 -8.66E-05
7 18.857 5.897 0.0678 -2.77E-05 0.0681 7.69E-05
8 9.321 2.91 0.0288 -5.79E-06 0.0265 -4.47E-04
9 7.505 2.332 0.0224 -4.56E-06 0.0223 -3.98E-06
T = 410 K
3 178.322 21.267 0.6593 -3.38E-04 0.6556 1.22E-03
4 133.144 19.333 0.5299 -1.62E-04 0.5299 -2.86E-04
7 19.542 5.897 0.0675 -2.75E-05 0.0683 -4.27E-05
8 9.564 2.91 0.0287 -5.53E-06 0.0287 8.86E-04
9 7.728 2.332 0.0223 -4.46E-06 0.0223 -3.01E-06
T = 420 K
3 184.939 21.267 0.656 -3.36E-04 0.6582 -6.86E-04
4 138.429 19.333 0.5283 -1.58E-04 0.5285 -5.11E-07
7 20.223 5.897 0.0673 -2.73E-05 N/A N/A
8 9.895 2.91 0.0287 -5.28E-06 0.0309 -4.53E-04
9 7.951 2.332 0.0223 -4.45E-06 0.0223 3.64E-06
103
Table 34. Continued
Isochore P' ρ*
Analytical Derivatives Numerical Derivatives
(dP/dT) (d2P/dT2) (dP/dT) (d2P/dT2)
(MPa) (kmol/m3) (MPa/K) (MPa/K2) (MPa/K) (MPa/K2)
T = 430 K
3 191.487 21.267 0.6526 -3.33E-04 0.6562 2.73E-04
4 143.714 19.333 0.5267 -1.54E-04 0.5272 -2.54E-04
8 10.182 2.91 0.0286 -5.05E-06 0.0286 1.56E-07
9 8.175 2.332 0.0222 -4.43E-06 0.0223 -1.43E-05
T = 440 K
3 198.062 21.267 0.6493 -3.31E-04 0.6535 -8.14E-04
4 148.974 19.333 0.5252 -1.51E-04 0.5246 -2.84E-04
8 10.468 2.91 0.0286 -4.82E-06 0.0286 -5.43E-06
9 8.396 2.332 0.0222 -4.42E-06 0.0222 -1.12E-06
T = 450 K
3 204.556 21.267 0.646 -3.29E-04 N/A N/A
4 154.205 19.333 0.5237 -1.48E-04 N/A N/A
8 10.754 2.91 0.0285 -4.60E-06 N/A N/A
9 8.618 2.332 0.0221 -4.41E-06 0.022 -9.61E-07
104
APPENDIX E. EXPRESSION OF RESIDUAL HEAT CAPACITY
The heat capacity at constant volume (Eq. 41) and the residual heat capacity at
constant volume (Eq. 42):
v
v
UC
T
(41)
rr
v
v
UC
T
(42)
The residual internal energy, Helmholtz energy and entropy are defined as:
r r rU A S
RT RT R (43)
0
1rA d
ZRT
(44)
0
11
rS P d
R R T
(45)
0
1r P dU P T
T
(46)
Replacing Eqs. 44 and 45 into Eq. 43, and taking the derivative with respect to
temperature:
105
2
2 2 2
0 01/
rU P d P P dT
T T T T
(47)
Replacing Eq. 47 into Eq. 42:
2
2 2
0
r
v
P dC T
T
(48)
106
APPENDIX F. REGRESSION RESULTS OF INTEGRAND FUNCTION
Table 35. Regression results of Eq. 35 for set 1
Coefficient Standard
Error Lower 95% Upper 95% rms
T = 220 K
c1 -2.56E-08 1.13E-09 -3.56E-08 -1.56E-08
2.86E-13 c2 7.33E-07 2.46E-08 4.73E-07 9.94E-07
c3 -4.39E-06 3.81E-07 -5.54E-06 -3.23E-06
T = 230 K
c1 -1.71E-08 1.66E-09 -3.12E-08 -3.00E-09
6.61E-13 c2 4.95E-07 3.57E-08 1.24E-07 8.66E-07
c3 -3.43E-06 5.26E-07 -5.09E-06 -1.76E-06
T = 240 K
c1 -1.57E-08 1.14E-09 -2.54E-08 -6.06E-09
3.11E-13 c2 4.36E-07 2.45E-08 1.81E-07 6.90E-07
c3 -2.83E-06 3.61E-07 -3.97E-06 -1.68E-06
T = 250 K
c1 -1.43E-08 9.57E-10 -2.24E-08 -6.15E-09
2.20E-13 c2 3.90E-07 2.06E-08 1.75E-07 6.04E-07
c3 -2.53E-06 3.04E-07 -3.49E-06 -1.57E-06
T = 260 K
c1 -1.33E-08 8.34E-10 -2.04E-08 -6.21E-09
1.67E-13 c2 3.59E-07 1.80E-08 1.72E-07 5.45E-07
c3 -2.34E-06 2.65E-07 -3.18E-06 -1.51E-06
T = 270 K
c1 -1.26E-08 7.48E-10 -1.89E-08 -6.23E-09
1.34E-13 c2 3.37E-07 1.61E-08 1.69E-07 5.04E-07
c3 -2.21E-06 2.37E-07 -2.96E-06 -1.46E-06
107
Table 35. Continued
Coefficient Standard
Error Lower 95% Upper 95% rms
T = 280 K
c1 -1.30E-08 4.25E-10 -1.66E-08 -9.41E-09
4.33E-14 c2 3.53E-07 9.14E-09 2.58E-07 4.48E-07
c3 -2.36E-06 1.35E-07 -2.78E-06 -1.93E-06
T = 290 K
c1 -1.16E-08 6.42E-10 -1.71E-08 -6.16E-09
9.90E-14 c2 3.07E-07 1.38E-08 1.63E-07 4.51E-07
c3 -2.03E-06 2.04E-07 -2.67E-06 -1.39E-06
T = 300 K
c1 -1.13E-08 6.10E-10 -1.64E-08 -6.08E-09
8.94E-14 c2 2.97E-07 1.31E-08 1.60E-07 4.33E-07
c3 -1.96E-06 1.94E-07 -2.58E-06 -1.35E-06
Table 36. Regression results of Eq. 35 for set 2
Coefficient Standard
Error Lower 95% Upper 95% rms
T = 310 K
c1 -8.52E-09 9.32E-10 -2.25E-08 5.44E-09
1.16E-13 c2 2.26E-07 1.95E-08 -1.13E-07 5.65E-07
c3 -1.77E-06 2.75E-07 -2.81E-06 -7.23E-07
T = 320 K
c1 -8.42E-09 8.98E-10 -2.19E-08 5.05E-09
1.08E-13 c2 2.23E-07 1.88E-08 -1.04E-07 5.50E-07
c3 -1.73E-06 2.65E-07 -2.74E-06 -7.24E-07
T = 330 K
c1 -8.34E-09 8.75E-10 -2.15E-08 4.79E-09
1.03E-13 c2 2.20E-07 1.83E-08 -9.83E-08 5.39E-07
c3 -1.70E-06 2.58E-07 -2.68E-06 -7.18E-07
108
Table 36. Continued
Coefficient Standard
Error Lower 95% Upper 95% rms
T = 340 K
c1 -8.28E-09 8.61E-10 -2.12E-08 4.63E-09
9.94E-14 c2 2.18E-07 1.80E-08 -9.53E-08 5.31E-07
c3 -1.67E-06 2.54E-07 -2.64E-06 -7.06E-07
T = 350 K
c1 -8.22E-09 8.53E-10 -2.10E-08 4.57E-09
9.76E-14 c2 2.16E-07 1.79E-08 -9.42E-08 5.27E-07
c3 -1.65E-06 2.52E-07 -2.60E-06 -6.90E-07
T = 360 K
c1 -8.18E-09 8.50E-10 -2.09E-08 4.57E-09
9.69E-14 c2 2.15E-07 1.78E-08 -9.48E-08 5.24E-07
c3 -1.62E-06 2.51E-07 -2.58E-06 -6.71E-07
T = 370 K
c1 -8.13E-09 8.51E-10 -2.09E-08 4.62E-09
9.71E-14 c2 2.13E-07 1.78E-08 -9.65E-08 5.23E-07
c3 -1.60E-06 2.51E-07 -2.56E-06 -6.49E-07
T = 380 K
c1 -3.72E-09 4.68E-10 -1.20E-08 4.51E-09
2.41E-14 c2 9.41E-08 9.81E-09 -1.11E-07 2.99E-07
c3 -1.07E-06 1.49E-07 -1.77E-06 -3.55E-07
T = 390 K
c1 -3.70E-09 4.90E-10 -1.23E-08 4.92E-09
2.65E-14 c2 9.31E-08 1.03E-08 -1.21E-07 3.08E-07
c3 -1.05E-06 1.56E-07 -1.79E-06 -3.05E-07
109
Table 37. Regression results of Eq. 35 for set 3
Coefficient Standard
Error Lower 95% Upper 95% rms
T = 400 K
c1 -1.70E-09 8.64E-10 -1.74E-08 1.40E-08
3.03E-14 c2 4.96E-08 1.72E-08 -3.25E-07 4.25E-07
c3 -9.18E-07 2.27E-07 -2.15E-06 3.09E-07
T = 410 K
c1 -1.51E-09 9.12E-10 -1.81E-08 1.51E-08
3.37E-14 c2 4.43E-08 1.81E-08 -3.51E-07 4.40E-07
c3 -8.83E-07 2.40E-07 -2.18E-06 4.12E-07
T = 420 K
c1 -1.46E-09 9.64E-10 -1.90E-08 1.61E-08
3.77E-14 c2 4.29E-08 1.92E-08 -3.76E-07 4.61E-07
c3 -8.66E-07 2.53E-07 -2.24E-06 5.03E-07
T = 430 K
c1 -1.08E-08 8.23E-10 -4.96E-08 2.79E-08
5.85E-15 c2 2.57E-07 1.67E-08 -6.38E-07 1.15E-06
c3 -1.30E-06 2.42E-07 -3.55E-06 9.45E-07
T = 440 K
c1 -1.11E-08 9.98E-10 -5.81E-08 3.58E-08
8.60E-15 c2 2.63E-07 2.02E-08 -8.21E-07 1.35E-06
c3 -1.30E-06 2.93E-07 -4.03E-06 1.42E-06
T = 450 K
c1 -1.14E-08 1.17E-09 -6.63E-08 4.35E-08
1.18E-14 c2 2.68E-07 2.36E-08 -1.00E-06 1.54E-06
c3 -1.30E-06 3.42E-07 -4.49E-06 1.88E-06
110
APPENDIX G. IDEAL GAS HEAT CAPACITY DIPPR PROJECT
2 2
/ /
sinh / cosh /
ig
v
C T E TC A B D
C T E T
(49)
The ideal gas heat capacity (Cv ig) calculated with Eq. 49 has units of J/kmol-K.
Temperature range: 298.1 – 1500 K (Methane) and 50 – 1500 K (Ethane and Propane) and
maximum percentage of deviation: 0.1% (Methane) and 0.3% (Ethane and Propane). The
parameters needed in Eq. 49 appear in Table G-1 for each substance.
Table 38. Parameters for Eq. 49
Parameter Methane Ethane Propane
A 33298 44256 59474
B 79933 84737 126610
C 2086.9 872.24 844.31
D 41602 67130 86165
E 991.96 2430.4 2482.7
111
APPENDIX H. RESIDUAL AND ABSOLUTE HEAT CAPACITY
Table 39. Residual and absolute heat capacity for set 1
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 220 K
1 112.942 24.817 -1.74E-06 2.907 28.992 32.971
2 66.321 22.664 -9.83E-07 2.249 28.333 31.764
3 46.929 21.267 -4.63E-07 2.06 28.145 31.167
5 12.041 14.699 6.98E-07 2.704 28.789 31.115
6 8.08 10.242 5.69E-07 3.433 29.517 33.076
8 4.004 2.91 -1.51E-06 2.171 28.255 29.219
9 3.432 2.332 -3.28E-06 1.835 27.92 28.578
10 2.543 1.63 -3.69E-06 1.367 27.451 27.791
T = 230 K
1 124.133 24.817 -1.70E-06 4.502 30.782 33.101
2 74.941 22.664 -1.01E-06 3.855 30.136 31.9
3 54.138 21.267 -5.81E-07 3.6 29.88 31.292
5 15.007 14.699 2.59E-07 3.438 29.718 30.763
6 9.629 10.242 1.60E-07 3.504 29.784 31.899
7 6.918 5.897 -2.20E-06 2.935 29.216 31.219
8 4.31 2.91 -1.44E-06 1.843 28.124 28.962
9 3.641 2.332 -1.30E-06 1.545 27.825 28.427
10 2.702 1.63 -3.69E-06 1.139 27.42 27.761
T = 240 K
1 135.191 24.817 -1.67E-06 3.839 30.346 33.264
2 83.506 22.664 -1.02E-06 3.144 29.651 32.071
3 61.316 21.267 -6.53E-07 2.863 29.37 31.455
5 17.975 14.699 8.54E-08 2.662 29.168 30.601
6 11.186 10.242 7.83E-08 2.81 29.317 31.308
7 7.653 5.897 -1.07E-06 2.438 28.944 30.702
8 4.618 2.91 -1.38E-06 1.561 28.068 28.832
9 3.868 2.332 -1.01E-06 1.313 27.82 28.378
10 2.856 1.63 -3.10E-06 0.972 27.479 27.807
112
Table 39. Continued
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 250 K
1 146.157 24.817 -1.64E-06 3.907 30.671 33.46
2 92.003 22.664 -1.03E-06 3.187 29.951 32.277
3 68.488 21.267 -7.00E-07 2.884 29.648 31.656
5 20.947 14.699 3.44E-09 2.563 29.327 30.566
6 12.75 10.242 5.22E-08 2.654 29.418 30.985
7 8.372 5.897 -9.33E-07 2.284 29.048 30.397
8 4.918 2.91 -1.32E-06 1.459 28.223 28.805
9 4.099 2.332 -9.26E-07 1.227 27.991 28.413
10 3.014 1.63 -2.72E-06 0.908 27.672 27.918
T = 260 K
1 157.011 24.817 -1.62E-06 4.018 31.072 33.69
2 100.439 22.664 -1.03E-06 3.276 30.33 32.517
3 75.309 21.267 -7.30E-07 2.956 30.009 31.894
5 23.921 14.699 -4.01E-08 2.544 29.598 30.626
6 14.318 10.242 4.13E-08 2.59 29.643 30.826
7 9.092 5.897 -8.94E-07 2.209 29.263 30.242
8 5.218 2.91 -1.27E-06 1.407 28.461 28.863
9 4.331 2.332 -8.93E-07 1.183 28.236 28.522
10 3.166 1.63 -2.46E-06 0.875 27.928 28.087
T = 270 K
1 168.125 24.817 -1.59E-06 4.129 31.504 33.953
2 108.796 22.664 -1.03E-06 3.365 30.74 32.791
3 83.088 21.267 -7.49E-07 3.031 30.405 32.167
5 26.888 14.699 -6.54E-08 2.551 29.926 30.762
6 15.876 10.242 3.60E-08 2.563 29.938 30.788
7 9.819 5.897 -8.76E-07 2.172 29.546 30.206
8 5.525 2.91 -1.21E-06 1.38 28.755 28.995
9 4.564 2.332 -8.76E-07 1.159 28.534 28.694
10 3.32 1.63 -2.28E-06 0.857 28.232 28.308
113
Table 39. Continued
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 280 K
1 178.88 24.817 -1.57E-06 4.474 32.201 34.249
2 117.143 22.664 -1.02E-06 3.685 31.411 33.098
3 90.161 21.267 -7.62E-07 3.342 31.068 32.475
5 29.861 14.699 -8.10E-08 2.867 30.594 30.962
6 17.442 10.242 3.31E-08 2.873 30.6 30.847
7 10.517 5.897 -8.65E-07 2.42 30.147 30.267
8 5.817 2.91 -1.16E-06 1.531 29.258 29.19
9 4.793 2.332 -1.45E-06 1.285 29.012 28.922
10 3.475 1.63 -2.14E-06 0.949 28.676 28.576
T = 290 K
1 189.51 24.817 -1.55E-06 4.337 32.446 34.576
2 125.417 22.664 -1.02E-06 3.533 31.642 33.437
3 97.209 21.267 -7.70E-07 3.173 31.282 32.815
5 32.832 14.699 -9.11E-08 2.596 30.705 31.217
6 19.007 10.242 3.14E-08 2.564 30.673 30.987
7 11.228 5.897 -8.57E-07 2.153 30.262 30.409
8 6.109 2.91 -1.11E-06 1.364 29.473 29.441
9 5.02 2.332 -8.59E-07 1.145 29.254 29.199
10 3.626 1.63 -2.04E-06 0.846 28.955 28.887
T = 300 K
1 199.871 24.817 -1.53E-06 4.434 32.955 34.933
2 133.775 22.664 -1.01E-06 3.612 32.132 33.806
3 104.212 21.267 -7.74E-07 3.241 31.761 33.185
5 35.793 14.699 -9.80E-08 2.626 31.146 31.52
6 20.567 10.242 3.04E-08 2.579 31.099 31.197
7 11.925 5.897 -8.50E-07 2.16 30.68 30.621
8 6.402 2.91 -1.06E-06 1.366 29.886 29.74
9 5.247 2.332 -8.54E-07 1.147 29.667 29.521
10 3.777 1.63 -1.96E-06 0.848 29.368 29.237
114
Table 40. Residual and absolute heat capacity for set 2
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 310 K
2 141.875 22.664 -1.00E-06 4.654 33.612 34.203
3 110.777 21.267 -7.76E-07 4.26 33.218 33.585
4 79.253 19.333 -6.31E-07 3.847 32.806 32.879
7 12.645 5.897 -8.43E-07 2.193 31.151 30.891
8 6.701 2.91 -1.02E-06 1.32 30.278 30.083
9 5.477 2.332 -8.49E-07 1.099 30.057 29.883
10 3.929 1.63 -1.90E-06 0.804 29.763 29.622
T = 320 K
2 149.998 22.664 -9.95E-07 4.689 34.111 34.626
3 117.681 21.267 -7.77E-07 4.288 33.71 34.011
4 84.729 19.333 -5.99E-07 3.868 33.291 33.302
7 13.35 5.897 -8.38E-07 2.211 31.633 31.212
8 6.995 2.91 -9.74E-07 1.332 30.755 30.465
9 5.704 2.332 -8.46E-07 1.11 30.532 30.28
10 4.076 1.63 -1.85E-06 0.812 30.235 30.039
T = 330 K
2 158.073 22.664 -9.88E-07 4.724 34.634 35.073
3 124.567 21.267 -7.76E-07 4.316 34.225 34.462
4 90.19 19.333 -5.71E-07 3.89 33.799 33.752
7 14.056 5.897 -8.32E-07 2.232 32.142 31.577
8 7.288 2.91 -9.33E-07 1.347 31.257 30.881
9 5.931 2.332 -8.42E-07 1.122 31.032 30.71
10 4.208 1.63 -1.81E-06 0.822 30.731 30.486
T = 340 K
2 166.119 22.664 -9.79E-07 4.76 35.178 35.543
3 131.394 21.267 -7.74E-07 4.344 34.762 34.935
4 95.626 19.333 -5.47E-07 3.911 34.33 34.225
7 14.756 5.897 -8.27E-07 2.256 32.674 31.981
8 7.58 2.91 -8.93E-07 1.363 31.782 31.327
9 6.157 2.332 -8.39E-07 1.136 31.555 31.167
10 4.381 1.63 -1.78E-06 0.832 31.251 30.959
115
Table 40. Continued
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 350 K
2 174.109 22.664 -9.71E-07 4.795 35.742 36.033
3 138.193 21.267 -7.71E-07 4.372 35.319 35.429
4 101.041 19.333 -5.25E-07 3.933 34.88 34.719
7 15.446 5.897 -8.21E-07 2.28 33.227 32.417
8 7.871 2.91 -8.54E-07 1.381 32.328 31.8
9 6.383 2.332 -8.36E-07 1.151 32.098 31.651
10 4.534 1.63 -1.75E-06 0.843 31.79 31.455
T = 360 K
2 182.052 22.664 -9.63E-07 4.831 36.324 36.541
3 144.973 21.267 -7.68E-07 4.401 35.894 35.941
4 106.435 19.333 -5.05E-07 3.956 35.449 35.232
7 16.127 5.897 -8.16E-07 2.306 33.799 32.882
8 8.131 2.91 -8.17E-07 1.399 32.892 32.297
9 6.609 2.332 -8.33E-07 1.166 32.659 32.156
10 4.685 1.63 -1.72E-06 0.855 32.348 31.972
T = 370 K
2 189.943 22.664 -9.55E-07 4.867 36.922 37.065
3 151.746 21.267 -7.64E-07 4.43 36.485 36.47
4 111.816 19.333 -4.88E-07 3.979 36.034 35.762
7 16.815 5.897 -8.11E-07 2.333 34.388 33.372
8 8.455 2.91 -7.82E-07 1.417 33.472 32.815
9 6.833 2.332 -8.30E-07 1.182 33.237 32.682
10 4.828 1.63 -1.70E-06 0.867 32.922 32.508
T = 380 K
2 197.792 22.664 -9.46E-07 5.481 38.112 37.604
3 158.47 21.267 -7.60E-07 5.059 37.69 37.012
4 117.171 19.333 -4.72E-07 4.551 37.182 36.307
7 17.498 5.897 -8.06E-07 1.862 34.492 33.884
8 8.744 2.91 -7.48E-07 1.038 33.669 33.351
9 7.058 2.332 -8.28E-07 0.853 33.484 33.225
116
Table 40. Continued
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 390 K
2 205.593 22.664 -9.38E-07 5.481 38.7 38.155
3 165.166 21.267 -7.56E-07 5.059 38.278 37.568
4 122.508 19.333 -4.58E-07 4.551 37.77 36.864
7 18.18 5.897 -8.01E-07 1.862 35.08 34.415
8 9.033 2.91 -7.15E-07 1.038 34.257 33.904
9 7.282 2.332 -8.25E-07 0.853 34.072 33.784
Table 41. Residual and absolute heat capacity for set 3
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 400 K
3 171.827 21.267 -7.51E-07 5.51 39.328 38.134
4 127.83 19.333 -4.45E-07 5.036 38.853 37.434
7 18.857 5.897 -7.96E-07 1.868 35.685 34.963
8 9.321 2.91 -6.84E-07 0.99 34.808 34.47
9 7.505 2.332 -8.39E-07 0.806 34.623 34.355
T = 410 K
3 178.322 21.267 -7.47E-07 5.579 40.004 38.71
4 133.144 19.333 -4.34E-07 5.098 39.523 38.012
7 19.542 5.897 -7.91E-07 1.862 36.287 35.523
8 9.564 2.91 -6.53E-07 0.982 35.407 35.048
9 7.728 2.332 -8.20E-07 0.798 35.223 34.938
T = 420 K
3 184.939 21.267 -7.42E-07 5.636 40.676 39.294
4 138.429 19.333 -4.23E-07 5.149 40.19 38.6
7 20.223 5.897 -7.86E-07 1.874 36.915 36.096
8 9.895 2.91 -6.24E-07 0.987 36.028 35.636
9 7.951 2.332 -8.17E-07 0.802 35.843 35.531
117
Table 41. Continued
Isochore P' ρ* (d2P/dT2)·ρ-2 Cv
r Cv Cv (GERG
EoS)
(MPa) (kmol/m3) (MPa·m6/K2·kmol2) (kJ/kmol-K)
T = 430 K
3 191.487 21.267 -7.37E-07 1.924 37.587 39.885
4 143.714 19.333 -4.13E-07 1.453 37.115 39.193
8 10.182 2.91 -5.96E-07 1.203 36.866 36.232
9 8.175 2.332 -8.15E-07 1.028 36.69 36.132
T = 440 K
3 198.062 21.267 -7.32E-07 1.756 38.046 40.481
4 148.974 19.333 -4.04E-07 1.28 37.57 39.793
8 10.468 2.91 -5.69E-07 1.22 37.51 36.836
9 8.396 2.332 -8.12E-07 1.045 37.334 36.739
T = 450 K
3 204.556 21.267 -7.27E-07 1.594 38.514 41.081
4 154.205 19.333 -3.96E-07 1.113 38.034 40.397
8 10.754 2.91 -5.43E-07 1.237 38.158 37.446
9 8.618 2.332 -8.10E-07 1.061 37.981 37.352
118
APPENDIX I. INTEGRAND FUNCTION OF NUMERICAL DERIVATIVES
Figure 55. Integrand function of Eq. 34 for subset 1 using numerical derivatives
Figure 56. Integrand function of Eq. 34 for subset 2 using numerical derivatives
-5.0E-05
-3.0E-05
-1.0E-05
1.0E-05
3.0E-05
5.0E-05
0 5 10 15 20 25 30
(d2P
/dT
2)/
ρ2
ρ (kmol/m3)
Numerical 220 K to 300 K
300
220
230
240
250
260
270
280
290
-2.0E-04
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25
(d2P
/dT
2)/
ρ2
ρ (kmol/m3)
Numerical 310 K to 390 K
310
320
330
340
350
360
370
380
119
Figure 57. Integrand function of Eq. 34 for subset 3 using numerical derivatives
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
0 5 10 15 20 25
(d2P
/dT
2)/
ρ2
ρ (kmol/m3)
Numerical 400 to 450 K
400
410
420
430
440
450
120
APPENDIX J. MATLAB CODE
Code for fitting truly isochoric data and determining numerical and analytical
derivatives. Example (Isochore 3):
function ISO31 clc clear format long x_all=[]; residual_all=[]; ppredicted_all=[]; s_all = [] devCoef_all=[] first_derivative=[]; second_derivative=[]; dp_all = []; dpp_all = []; dev_all = [];
% p = xlsread('isochore.xlsx', 3, 'L6:L37') Y = xlsread('isochore.xlsx', 3, 'F6:F37') p = xlsread('isochore.xlsx', 3, 'AJ6:AJ30') T = xlsread('isochore.xlsx', 3, 'AI6:AI30')
h = xlsread('isochore.xlsx', 3, 'S6:S34') w = xlsread('isochore.xlsx', 3, 'P6:P35')
%numerical derivatives %w number of data for numerical derivatives
n = size (w)
for i = 2:(n(1)-1)
c = w(i); a = w(i+1); b = w(i-1); l = h(i);
di = ((a)-(b))./(2.*l); dii = ((a)-2.*(c)+(b))/(l.^(2));
first_derivative=[first_derivative;di]; second_derivative=[second_derivative;dii];
121
end
x0 = [1,1,1,1,1,1]; % Starting guess at the solution
options=optimset('TolFun',1E-20,'TolX',1E-
20,'MaxFunEvals',40000,'MaxIter',40000,'Algorithm','levenberg-
marquardt' );
[x,resnorm,residual] =
lsqnonlin(@functiongoal,x0,[],[],options,p,T); % Invoke optimizer
hold on plot(T, residual,'or') ppredicted = (x(1)+x(2).*T+x(3).*T.^2)./(x(4)+x(5).*T+x(6).*T.^2) dp = (((-x(1)-x(2).*Y-
x(3).*Y.^2).*(x(5)+2.*x(6).*Y))+((x(4)+x(5).*Y+x(6).*Y.^2).*(x(2)
+2.*x(3).*Y)))./((x(4)+x(5).*Y+x(6).*Y.^2).^2); dpp = (((x(4)+x(5).*Y+x(6).*Y.^2).^2).*(2.*x(6).*(-x(1)-x(2).*Y-
x(3).*Y.^2)+2.*x(3).*(x(4)+x(5).*Y+x(6).*Y.^2))-
((2.*(x(5)+2.*x(6).*Y).*(x(4)+x(5).*Y+x(6).*Y.^2)).*((x(5)+2.*x(6
).*Y).*(-x(1)-x(2).*Y-
x(3).*Y.^2)+(x(4)+x(5).*Y+x(6).*Y.^2).*(x(2)+2.*x(3).*Y))))./((x(
4)+x(5).*Y+x(6).*Y.^2).^4);
dev_fit = ((p-ppredicted)./p).*100
% x_all is a a variable that save all the coefficients
x_all=[x_all;x]; residual_all=[residual_all;residual]; ppredicted_all = [ppredicted_all;ppredicted]; dp_all = [dp_all;dp]; dpp_all = [dpp_all;dpp]; dev_all = [dev_all,dev_fit];
disp('Coefficients values') disp(x) disp('Resnorm=') disp(resnorm)
%Coefficients errors n=size(p); m=size(x0);
rms=sum((ppredicted-p).^2)./(n(1)-m(2));
% this derivative changes with the form of the fit
122
dX=[ 1./(x(4)+x(5).*T+x(6).*T.^2), T./(x(4)+x(5).*T+x(6).*T.^2),
T.^2./(x(4)+x(5).*T+x(6).*T.^2),(-x(1)-x(2).*T-
x(3).*T.^2)./(x(4)^2),(-x(1)-x(2).*T-
x(3).*T.^2)./(T.*(x(5)^2)),(-x(1)-x(2).*T-
x(3).*T.^2)./(T.^2.*(x(6)^2))];
%Matrix A for k=1:m(2) for ii=1:n(1) A(ii,k)=dX(ii,k); At(k,ii)=dX(ii,k); end end
E=(At*A).^(-1);
for k=1:m(2) s(k)=(rms*E(k,k)).^(1/2); devCoef(k)=(tinv(0.95,(n(1)-m(2))))*s(k); end
s_all = [s_all;s]; devCoef_all=[devCoef_all;devCoef];
save ISO31
function f = functiongoal(x,p,T) f = (p)-((x(1)+x(2).*T+x(3).*T.^2)./(x(4)+x(5).*T+x(6).*T.^2));
Code for determining residual and absolute heat capacity (Example: T = 280K)
function Temp280c clc clear format long x_all=[]; residual_all=[]; ypredicted_all=[]; integral_all=[]; s_all = [] devCoef_all=[] Heatcapacity=[]; confidence=[]; P11=[];
123
P12=[];
y = xlsread('isotherm.xlsx', 7, 'K4:K12') r = xlsread('isotherm.xlsx', 7, 'F4:F12') T = xlsread('isotherm.xlsx', 7, 'C4:C12')
fitresults = fit(r,y,'poly2') ci = confint(fitresults,0.95) confidence=[confidence;ci];
p11= predint(fitresults,r,0.95,'observation','off'); p12= predint(fitresults,r,0.95,'function','on'); P11=[P11,p11]; P12=[P12,p12];
plot(fitresults,r,y),hold on, plot(r,p12); x0 = [0,0,0]; % Starting guess at the solution
options=optimset('TolFun',1E-20,'TolX',1E-
20,'MaxFunEvals',40000,'MaxIter',40000,'Algorithm','levenberg-
marquardt' );
[x,resnorm,residual] =
lsqnonlin(@functiongoal,x0,[],[],options,y,r); % Invoke optimizer
hold on plot(r, residual,'or') ypredicted = x(1).*r.^2+x(2).*r+x(3); integral = x(3).*r+(x(2)/2).*r.^2+(x(1)/3).*r.^3
% x_all is a a variable that save all the coefficients
x_all=[x_all;x]; residual_all=[residual_all;residual]; ypredicted_all = [ypredicted_all;ypredicted]; integral_all = [integral_all;integral];
disp('Coefficients values') disp(x) disp('Resnorm=') disp(resnorm)
%Cv residual calculation n = size (T)
for i = 1:n(1)
124
Cv = T(i).*-1000.*integral(i);
Heatcapacity=[Heatcapacity;Cv];
end
%Coefficients errors n=size(y); m=size(x0);
rms=sum((ypredicted-y).^2)./(n(1)-m(2));
% this derivative changes with the form of the fit dX=[r.^2, r, ones(n,1)];
%Matrix A for k=1:m(2) for ii=1:n(1) A(ii,k)=dX(ii,k); At(k,ii)=dX(ii,k); end end
E=(At*A).^(-1);
for k=1:m(2) s(k)=(rms*E(k,k)).^(1/2); devCoef(k)=(tinv(0.95,(n(1)-m(2))))*s(k); end
s_all = [s_all;s]; devCoef_all=[devCoef_all;devCoef];
save Temp280c
function f = functiongoal(x,y,r) f = (y)-(x(1).*r.^2+x(2).*r+x(3));